#7625110

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1232 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26)(19,28,30,24)(20,34,27,35)(21,37,38,25)(23,40,39,36)(29,44,42,48)(31,41,33,43)(32,46,47,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,29)(21,33)(22,30)(23,32)(25,43)(26,28)(27,42)(31,38)(34,48)(35,44)(36,46)(37,41)(39,47)(40,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,35,18,21,36,23,25,22,27,34,26,38,40,39,37)(19,29,44,24,33,46,32,43,30,42,48,28,31,45,47,41);(3,12)(7,10)(11,13)(15,16)(19,30)(24,28)(29,42)(31,33)(32,47)(41,43)(44,48)(45,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,22,38)(18,25,26,37)(19,31,30,33)(20,36,27,40)(23,34,39,35)(24,41,28,43)(29,45,42,46)(32,44,47,48)

Polycyclic presentation:

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

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, 7532306);