#6485188

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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)(11,15)(12,16)(13,14)(17,18)(19,29)(20,25)(21,26)(22,36)(23,27)(24,32)(28,38)(30,45)(31,46)(33,44)(34,43)(35,42)(37,41)(39,47)(40,48);(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,24,27,41)(20,31,21,30)(22,37,38,32)(25,40,26,39)(28,44,36,29)(34,47,35,48)(42,45,43,46);(2,5)(6,8)(10,14)(15,16)(17,20)(18,25)(19,30)(21,23)(22,34)(24,39)(26,27)(28,42)(29,45)(31,33)(32,47)(35,38)(36,43)(37,48)(40,41)(44,46);(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,27,25)(19,31,33,30)(22,35,38,34)(24,40,41,39)(28,43,36,42)(29,46,44,45)(32,48,37,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,27)(20,21)(24,41)(28,36)(29,44)(30,31)(34,35)(47,48)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=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, 6392384);