#72303159

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11174
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
socle	C₂	C₂×#11174
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=h,c^2=j,d^2=eg,b^a=bf,d^a=dfhij,d^b=de,d^c=dj,e^a=ei,e^b=eij,f^d=fi,f^e=fj,g^a=gij,g^b=gij,g^f=gj,h^d=hij,h^e=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303177, #72303324, #72303445
maximal normal subgroup of	#72303177, #72303324, #72303445

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, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.5*F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);