#7601315

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=gi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7508511);