#7623223

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530419);