#7622772

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3481	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1327 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)(9,10,13,18,14,12,24,25,15,19,27,33,28,26,17,32)(11,16,22,31,23,21,37,38,20,29,35,39,36,34,30,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,30)(18,31)(19,29)(24,37)(25,38)(26,34)(27,35)(28,36)(32,40)(33,39);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,30)(18,31)(19,29)(24,37)(25,38)(26,34)(27,35)(28,36)(32,40)(33,39);(2,6)(5,7)(11,20)(16,29)(21,34)(22,35)(23,36)(30,37)(31,39)(38,40);(9,12,13,25,14,19,24,33,15,26,27,32,28,10,17,18)(11,21,22,38,23,29,37,39,20,34,35,40,36,16,30,31)

Polycyclic presentation:

2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315511, #72315864, #72315883
maximal normal subgroup of	#72315511, #72315864, #72315883

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529968);