#7624045

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

1347 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)(9,10,12,17)(11,20,21,15)(13,16,24,32)(14,27,25,19)(18,34,33,26)(22,28,37,31)(23,30,38,36)(29,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,12,21)(10,15,17,20)(13,22,24,37)(14,23,25,38)(16,31,32,28)(18,29,33,35)(19,30,27,36)(26,40,34,39);(10,16)(11,21)(13,26)(14,25)(15,28)(17,32)(18,19)(20,31)(22,39)(24,34)(27,33)(29,36)(30,35)(37,40);(2,5)(6,8)(11,21)(15,20)(22,37)(23,38)(28,31)(29,35)(30,36)(39,40);(1,3,4,7)(2,6,5,8)(11,21)(15,20)(22,37)(23,38)(28,31)(29,35)(30,36)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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	#72260392, #72286697, #72313684, …
maximal normal subgroup of	#72260392, #72286697, #72313684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531241);