#7623574

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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 ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(18,23)(20,30)(21,32)(24,35)(25,36)(26,39)(28,42)(29,41)(31,34)(33,45)(37,46)(38,47)(40,48)(43,44);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(18,24)(19,27)(23,35)(26,34)(28,40)(29,41)(31,39)(42,48);(3,10)(6,9)(11,16)(13,15)(17,18)(19,25)(20,31)(21,34)(22,24)(23,30)(26,32)(27,37)(28,43)(29,44)(33,39)(35,45)(36,42)(38,41)(40,47)(46,48);(2,7)(6,9)(8,14)(13,15)(17,19,22,27)(18,25,24,37)(20,28,33,40)(21,29,32,41)(23,36,35,46)(26,38,34,44)(30,42,45,48)(31,43,39,47);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,27)(25,37)(28,40)(29,41)(36,46)(38,44)(42,48)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530770);