#6485247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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

1411 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)(17,18,22,25)(19,27,29,42)(20,24,32,41)(21,34,36,26)(23,35,38,30)(28,43,31,45)(33,48,47,40)(37,46,39,44);(9,10)(11,16)(12,15)(13,14)(17,19)(18,23)(20,31)(21,35)(22,29)(24,39)(25,38)(26,42)(27,34)(28,32)(30,36)(33,44)(37,41)(40,43)(45,48)(46,47);(2,5)(6,8)(10,14)(15,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,43)(29,31)(30,44)(34,48)(35,46)(36,47)(38,39)(42,45);(9,11,13,12)(10,15,14,16)(19,29)(23,38)(27,42)(28,31)(30,35)(37,39)(43,45)(44,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,25)(20,32)(23,38)(26,34)(27,42)(28,31)(33,47)(44,46)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,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,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, 6392443);