#7624043

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286697, #72313684, #72315837, …
maximal normal subgroup of	#72286697, #72313684, #72315837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531239);