#7624205

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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

1545 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)(3,7)(4,6)(5,8)(10,17)(11,20)(13,16)(15,28)(18,33)(19,36)(21,38)(22,40)(23,39)(25,32)(26,27)(29,30)(31,34)(35,37);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,16)(13,24)(14,17)(15,19)(20,34)(21,30)(22,35)(23,31)(25,28)(26,36)(27,32)(29,37)(33,39)(38,40);(9,11,14,23)(10,18,17,31)(12,21,24,35)(13,22,16,30)(15,29,27,40)(19,37,32,38)(20,26,39,28)(25,34,36,33);(3,8)(5,7)(9,12)(10,13)(11,21)(14,24)(15,25)(16,17)(18,22)(19,26)(20,38)(23,35)(27,36)(28,32)(29,34)(30,31)(33,40)(37,39);(2,6)(5,7)(9,13,14,16)(10,12,17,24)(11,22,23,30)(15,26,27,28)(18,21,31,35)(19,25,32,36)(20,29,39,40)(33,37,34,38)

Polycyclic presentation:

2^9|c^2=i,e^2=i,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,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315254, #72315846, #72315864, …
maximal normal subgroup of	#72315254, #72315846, #72315864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531401);