#6468216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375412);