#6473474

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=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, 6380670);