#6468211

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

1644 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)(17,18,23,36,22,27,21,26)(19,28,35,24,30,43,31,39)(20,25,37,48,33,41,34,42)(29,44,47,38,32,46,45,40);(10,14)(11,12)(17,19)(18,24)(20,32)(21,35)(22,30)(23,31)(25,40)(26,43)(27,39)(28,36)(29,33)(34,45)(37,47)(38,41)(42,44)(46,48);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,37)(24,38)(26,42)(27,41)(28,44)(30,32)(31,45)(35,47)(36,48)(39,40)(43,46);(9,12,13,11)(10,16,14,15)(19,30)(24,39)(28,43)(29,32)(31,35)(38,40)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(20,33)(25,41)(29,32)(34,37)(38,40)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=hi,e^2=g,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, 6375407);