#7625162

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1237 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,25)(19,28,30,24)(20,33,26,34)(22,38,36,35)(23,39,37,27)(29,44,40,47)(31,45,46,48)(32,41,42,43);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,28)(26,40)(27,41)(33,47)(34,44)(35,45)(36,46)(37,42)(38,48)(39,43);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,34,18,23,35,22,27,21,26,33,25,37,38,36,39)(19,29,44,24,32,45,31,41,30,40,47,28,42,48,46,43);(3,12)(7,10)(11,13)(15,16)(19,30)(24,28)(29,40)(31,46)(32,42)(41,43)(44,47)(45,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(19,30)(20,26)(27,39)(31,46)(32,42)(35,38)(44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,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,e^b=ei,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532358);