#72307292

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#20662
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#360178
prime radical	C₂×#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2289 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=ghj,d^2=fghij,g^2=ij,i^2=j,b^a=bh,c^a=cf,c^b=cg,d^b=dfgi,d^c=dfj,e^d=ej,f^b=fj,g^b=gi,g^c=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72343942, #72343971
maximal normal subgroup of	#72307299, #72343942, #72343971

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