#7625004

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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

1423 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)(3,7)(9,10)(11,20)(12,17)(13,27)(14,26)(15,19)(16,24)(18,28)(21,33)(22,34)(23,35)(25,40)(29,30)(31,38)(32,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,26)(13,18)(14,17)(15,19)(20,24)(21,30)(22,31)(23,36)(25,32)(27,28)(29,33)(34,38)(35,39)(37,40);(9,11,21,37,14,25,13,24,15,22,28,39,12,23,29,38)(10,16,30,40,17,32,18,20,19,31,27,35,26,36,33,34);(2,5)(6,8)(11,22)(16,31)(20,34)(23,25)(24,38)(32,36)(35,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12,15,14)(10,17,19,26)(11,23,22,25)(13,21,29,28)(16,32,31,36)(18,27,33,30)(20,35,34,40)(24,37,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318881, #72320442, #72320623
maximal normal subgroup of	#72318881, #72320442, #72320623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532200);