#7623782

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

1557 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 7530978);