#7625041

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624812
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532237);