#6529911

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	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 6437107);