#6561032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1314 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)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,36)(30,39)(31,41)(32,40)(35,42)(37,44)(38,43)(45,46)(47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,33,23,31,21,32)(18,24,26,29,28,40,27,41)(20,35,37,46,38,30,36,47)(25,42,43,48,44,39,34,45);(10,13)(14,16)(17,20,21,36,23,38,22,37)(18,25,27,34,28,44,26,43)(19,30,32,46,31,35,33,47)(24,39,41,48,40,42,29,45);(9,11)(10,14)(12,15)(13,16)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,38,37)(24,29,40,41)(25,43,44,34)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi

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, 6468228);