#72307227

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁵
socle	C₂	C₂×#18203
prime radical	#10577631	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(3,15,21,20)(4,22)(10,33,30,36)(11,31)(16,42,45,51)(18,43,23,49)(19,47,34,44)(24,26)(25,52)(35,58,37,59)(38,40)(39,57)(46,61,53,63)(48,54)(50,64,55,62)(56,60);(1,2,7,9)(3,16)(4,11,22,31)(5,14,8,12)(6,28,29,41)(10,34)(13,27,32,17)(15,42)(18,46)(19,30)(20,51)(21,45)(23,53)(24,40,26,38)(25,56,52,60)(33,44)(35,55)(36,47)(37,50)(39,54,57,48)(43,61)(49,63)(58,62)(59,64);(2,9)(10,30)(11,31)(12,14)(13,32)(16,45)(28,41)(33,36)(35,37)(38,40)(39,57)(42,51)(46,53)(56,60)(58,59)(61,63);(1,3,17,47,8,20,29,34,7,21,27,44,5,15,6,19)(2,10,28,42,14,36,32,45,9,30,41,51,12,33,13,16)(4,23,54,64,24,43,25,55,22,18,48,62,26,49,52,50)(11,37,60,61,38,58,39,53,31,35,56,63,40,59,57,46);(1,4)(2,11)(3,18)(5,26)(6,25)(7,22)(8,24)(9,31)(10,35)(12,40)(13,39)(14,38)(15,43)(16,46)(17,48)(19,50)(20,49)(21,23)(27,54)(28,56)(29,52)(30,37)(32,57)(33,58)(34,55)(36,59)(41,60)(42,61)(44,62)(45,53)(47,64)(51,63)

Polycyclic presentation:

2^10|a^2=hj,b^2=h,d^2=fgj,f^2=j,g^2=i,i^2=j,c^b=cj,d^a=df,d^b=dg,e^a=eh,e^d=ehj,f^b=fh,f^d=fh,f^e=fj,g^a=ghj,g^b=gi,g^d=gh,g^f=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307300, #72307386, #72307482
maximal normal subgroup of	#72307300, #72307386, #72307482

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8*F.10);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);