#7625034

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624802
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

1189 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)(21,35)(22,37)(23,27)(24,31)(26,29)(30,41)(32,47)(33,43)(34,40)(36,48)(38,45)(39,46)(42,44);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16);(17,19,29,44,22,32,36,46,23,33,34,45,20,30,21,31)(18,24,35,41,25,38,40,43,27,39,48,47,37,42,26,28);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,34,36,29)(24,38,39,42)(26,35,40,48)(28,41,43,47)(31,45,46,44)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532230);