#7623051

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	#60459	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1602 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)(9,10,14,19)(11,16,20,30)(12,15,25,29)(13,26,24,18)(17,33,28,23)(21,27,34,38)(22,37,35,32)(31,40,39,36);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(12,23)(13,24)(16,27)(17,18)(19,29)(21,36)(22,35)(25,33)(26,28)(30,38)(31,32)(34,40)(37,39);(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,31)(18,32)(19,30)(23,36)(24,35)(25,34)(26,37)(28,39)(29,38)(33,40);(11,20)(16,30)(21,34)(22,35)(27,38)(31,39)(32,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,c^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, 7530247);