#6442312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

1684 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,41)(28,40)(29,36)(30,44)(31,43)(32,37)(33,38)(34,39)(35,42)(45,48)(46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,30)(22,33)(24,39)(25,42)(29,34)(32,35)(36,41)(37,44)(40,43)(45,46);(10,14)(13,16)(17,19,21,29)(18,24,23,36)(20,31,32,28)(22,34,33,27)(25,43,37,40)(26,39,38,41)(30,46,35,45)(42,48,44,47);(17,20,21,32)(18,25,23,37)(19,28,29,31)(22,35,33,30)(24,40,36,43)(26,42,38,44)(27,45,34,46)(39,47,41,48);(1,3,4,7)(2,6,5,8)(19,29)(24,36)(27,34)(28,31)(39,41)(40,43)(45,46)(47,48)

Polycyclic presentation:

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

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, 6349508);