#7623123

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
prime radical	#33516	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1522 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)(9,10,15,20)(11,17,23,37)(12,18,14,30)(13,26,29,40)(16,31,34,22)(19,25,38,28)(21,35,24,36)(27,32,39,33);(2,5)(6,8);(9,11)(10,16)(12,21)(13,27)(14,24)(15,23)(17,26)(18,32)(19,36)(20,34)(22,28)(25,31)(29,39)(30,33)(35,38)(37,40);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,24)(13,22)(14,21)(15,23)(16,26)(18,36)(19,32)(20,37)(25,39)(27,28)(29,31)(30,35)(33,38)(34,40);(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,32,34,33)(17,35,37,36)(19,26,38,40)(22,27,31,39)

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=i,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, 7530319);