#7623853

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322867
maximal normal subgroup of	#72322867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531049);