#7625170

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

1234 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)(17,18,22,27)(19,29,30,24)(20,33,35,48)(21,39,40,26)(23,43,42,45)(25,44,32,37)(28,34,46,41)(31,38,47,36);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,30)(18,24,27,29)(20,28,35,46)(21,32,40,25)(23,31,42,47)(26,37,39,44)(33,41,48,34)(36,45,38,43);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,34,24,21,38,23,37,22,35,41,29,40,36,42,44)(18,25,45,47,26,19,28,33,27,32,43,31,39,30,46,48);(3,12)(7,10)(11,13)(15,16)(19,30)(20,35)(24,29)(25,32)(31,47)(33,48)(36,38)(37,44);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,22,40)(18,26,27,39)(19,25,30,32)(20,36,35,38)(23,41,42,34)(24,44,29,37)(28,43,46,45)(31,33,47,48)

Polycyclic presentation:

2^9|a^2=i,b^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,d^c=di,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, 7532366);