#6466723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1385 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,31)(21,23)(22,26)(24,30)(25,34)(28,36)(29,37)(32,39)(33,35)(38,43)(40,45)(41,44)(42,47)(46,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,32)(22,33)(24,36)(25,39)(27,38)(29,44)(30,43)(31,40)(34,45)(46,47);(9,12,13,11)(10,14,15,16)(17,19,22,28,21,30,33,43)(18,24,26,38,23,27,35,36)(20,29,32,42,34,44,45,48)(25,37,40,47,31,41,39,46);(17,20,33,45,21,34,22,32)(18,25,35,39,23,31,26,40)(19,29,43,48,30,44,28,42)(24,37,36,46,27,41,38,47);(1,3,4,7)(2,6,5,8)(20,34)(25,31)(29,44)(32,45)(37,41)(39,40)(42,48)(46,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373919);