#6529910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=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, 6437106);