#6466717

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

1381 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)(19,24)(20,30)(21,23)(22,26)(25,33)(27,38)(28,43)(29,34)(31,39)(32,35)(36,42)(37,44)(40,45)(41,46)(47,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,31)(22,32)(24,36)(25,39)(28,44)(29,42)(30,40)(33,45)(34,38)(46,47);(9,12,13,11)(10,14,15,16)(17,19,22,27,21,29,32,42)(18,24,26,38,23,34,35,36)(20,28,31,41,33,44,45,48)(25,37,40,47,30,43,39,46);(17,20,32,45,21,33,22,31)(18,25,35,39,23,30,26,40)(19,28,42,48,29,44,27,41)(24,37,36,46,34,43,38,47);(1,3,4,7)(2,6,5,8)(20,33)(25,30)(28,44)(31,45)(37,43)(39,40)(41,48)(46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,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, 6373913);