#6433883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1931 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(31,46)(34,40)(35,41)(37,44)(43,48)(45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,36,46)(32,42,40,48)(33,44,35,45);(11,15)(13,16)(17,20,22,33,23,34,21,35)(18,25,26,32,27,41,36,40)(19,28,31,44,29,43,30,45)(24,37,39,42,38,47,46,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,46)(42,48)(44,45);(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,34,35)(24,39,38,46)(25,40,41,32)(28,44,43,45)(37,48,47,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341079);