#6622824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,28,30,24)(20,32,34,41)(21,26,23,37)(25,36,42,33)(29,44,45,48)(31,40,35,39)(38,47,43,46);(2,6)(7,8)(10,14)(11,12)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,38)(26,40)(27,28)(32,48)(33,47)(34,45)(36,46)(37,39)(41,44)(42,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,21,36,22,34,23,33)(18,25,37,41,27,42,26,32)(19,29,35,46,30,45,31,47)(24,38,39,44,28,43,40,48);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,30)(24,28)(29,45)(31,35)(38,43)(39,40)(44,48)(46,47);(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,37)(19,31,30,35)(20,33,34,36)(24,39,28,40)(25,41,42,32)(29,46,45,47)(38,48,43,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

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, 6530020);