#6462141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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

1588 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)(17,18,22,24)(19,27,28,42)(20,25,32,39)(21,34,36,26)(23,35,38,30)(29,43,31,45)(33,48,47,41)(37,46,40,44);(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,35)(22,28)(24,38)(25,40)(26,42)(27,34)(29,32)(30,36)(33,44)(37,39)(41,43)(45,48)(46,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,24)(19,28)(25,39)(26,34)(29,31)(30,35)(41,48)(44,46);(10,14)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,39)(26,41)(27,43)(28,31)(30,44)(34,48)(35,46)(36,47)(38,40)(42,45);(1,3,4,7)(2,6,5,8)(20,32)(25,39)(29,31)(33,47)(37,40)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6369337);