#7623118

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1527 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)(3,6)(4,5)(7,8)(10,15)(11,19)(14,26)(16,29)(17,24)(18,25)(20,33)(21,32)(23,36)(27,28)(30,39)(31,35)(34,40)(37,38);(2,5)(6,8)(9,10,13,15)(11,16,21,28)(12,17,22,24)(14,18,25,26)(19,27,32,29)(20,30,34,38)(23,31,35,36)(33,37,40,39);(9,11)(10,16)(12,20)(13,21)(14,27)(15,28)(17,30)(18,32)(19,26)(22,34)(23,37)(24,38)(25,29)(31,40)(33,36)(35,39);(1,3,4,7)(2,6,5,8)(9,12,13,22)(10,17,15,24)(11,20,21,34)(14,23,25,35)(16,30,28,38)(18,31,26,36)(19,33,32,40)(27,37,29,39);(12,22)(17,24)(20,34)(23,35)(30,38)(31,36)(33,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530314);