#6599126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1738 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)(17,18,21,25)(19,27,29,23)(20,31,33,42)(22,36,35,26)(24,37,41,34)(28,44,32,48)(30,39,46,43)(38,45,40,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,39)(28,33)(31,44)(34,47)(35,46)(36,43)(37,45)(38,41)(42,48);(11,12)(14,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,41)(26,42)(27,40)(29,32)(30,47)(31,36)(34,35)(39,44)(43,48)(45,46);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(20,33)(24,41)(28,32)(31,42)(34,37)(38,40)(44,48)(45,47);(11,12)(14,16)(19,29)(20,33)(23,27)(24,41)(30,46)(31,42)(34,37)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506322);