#6485830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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

1371 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)(20,34,36,43)(21,28,22,27)(24,33,40,32)(25,37,44,38)(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)(20,35,37,47,36,30,38,46)(25,42,43,48,44,39,34,45);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,21,37,23,36,22,38)(18,25,27,43,26,44,28,34)(19,30,32,46,31,35,33,47)(24,39,29,45,40,42,41,48);(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)(20,36)(24,40)(27,28)(29,41)(30,35)(37,38)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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^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, 6393026);