#7502550

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(5,7)(6,8)(9,10,13,18)(11,20,22,34)(12,25,27,33)(14,29,15,19)(16,24,32,23)(17,28,26,30)(21,36,38,31)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,20)(12,26)(13,18)(14,19)(15,29)(16,24)(17,27)(21,37)(22,34)(23,32)(25,28)(30,33)(31,40)(35,38)(36,39);(9,11,13,22)(10,16,18,32)(12,21,27,38)(14,24,15,23)(17,31,26,36)(19,34,29,20)(25,37,33,35)(28,40,30,39);(2,7)(3,8)(9,12,13,27)(10,17,18,26)(11,21,22,38)(14,30,15,28)(16,31,32,36)(19,25,29,33)(20,35,34,37)(23,40,24,39);(1,4,5,6)(2,3,7,8)(12,27)(17,26)(21,38)(25,33)(28,30)(31,36)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409746);