#7625122

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

1273 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)(19,28,31,44)(20,24,33,27)(22,35,34,40)(23,36,38,32)(26,42,41,29)(30,43,47,45)(37,48,39,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,23)(20,27)(21,25)(22,26)(24,33)(28,32)(29,35)(30,39)(31,38)(34,41)(36,44)(37,47)(40,42)(43,48)(45,46);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,29,46,20,30,34,36,21,31,42,48,33,47,22,32)(18,23,35,45,27,39,41,44,25,38,40,43,24,37,26,28);(3,12)(7,10)(11,13)(15,16)(17,20,21,33)(18,24,25,27)(19,30,31,47)(22,29,34,42)(23,37,38,39)(26,40,41,35)(28,43,44,45)(32,46,36,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(23,38)(24,27)(26,41)(28,44)(35,40)(37,39)(43,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,d^2=i,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, 7532318);