#6485805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,26)(19,29,31,41)(20,34,38,44)(21,28,22,27)(24,33,40,32)(25,36,43,37)(30,39,35,42)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,21,32,23,31,22,33)(18,24,27,29,26,40,28,41)(20,35,36,47,38,30,37,46)(25,42,44,48,43,39,34,45);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,37,23,38,21,36)(18,25,28,34,26,43,27,44)(19,30,33,47,31,35,32,46)(24,39,41,48,40,42,29,45);(9,11,13,12)(10,15,14,16)(19,31)(24,40)(29,41)(30,35)(32,33)(39,42)(45,48)(46,47);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,26)(19,31)(25,43)(27,28)(30,35)(32,33)(34,44)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6393001);