#7625118

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

1244 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)(17,18,23,29)(19,24,33,43)(20,34,38,46)(21,26,40,28)(22,41,42,45)(25,44,31,37)(27,35,30,39)(32,47,48,36);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,30)(21,25)(22,32)(23,33)(26,44)(27,38)(28,37)(29,43)(31,40)(34,39)(35,46)(36,45)(41,47)(42,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,35,24,21,36,22,37,23,38,39,43,40,47,42,44)(18,25,45,32,28,33,27,34,29,31,41,48,26,19,30,46);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,21,23,40)(18,26,29,28)(19,31,33,25)(20,36,38,47)(22,39,42,35)(24,37,43,44)(27,45,30,41)(32,46,48,34)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,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

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, 7532314);