#7625132

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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
prime radical	#62440	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1241 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)(19,28,31,45)(20,24,34,27)(22,37,36,39)(23,38,41,32)(26,29,43,35)(30,44,33,48)(40,47,42,46);(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)(24,34)(28,32)(29,39)(30,42)(31,41)(33,40)(35,37)(36,43)(38,45)(44,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,29,46,20,33,22,32,21,31,35,47,34,30,36,38)(18,23,39,48,27,40,26,28,25,41,37,44,24,42,43,45);(3,12)(7,10)(11,13)(15,16)(17,20,21,34)(18,24,25,27)(19,30,31,33)(22,35,36,29)(23,40,41,42)(26,39,43,37)(28,44,45,48)(32,46,38,47);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(23,41)(24,27)(26,43)(28,45)(37,39)(40,42)(44,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532328);