#6463483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(11,16)(12,14)(13,15)(17,18,22,26)(19,27,31,43)(20,24,33,40)(21,35,28,25)(23,36,38,30)(29,42,32,46)(34,48,44,41)(37,47,39,45);(2,5)(6,8)(11,12)(14,16);(9,11,13,12)(10,14,15,16)(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,32,44,45,33,29,34,47)(24,39,48,42,40,37,41,46);(9,12,13,11)(10,14,15,16)(17,20,22,33)(18,24,26,40)(19,29,31,32)(21,34,28,44)(23,37,38,39)(25,41,35,48)(27,42,43,46)(30,45,36,47);(1,3,4,7)(2,6,5,8)(20,33)(24,40)(29,32)(34,44)(37,39)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6370679);