#6532514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2842 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:

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

Polycyclic presentation:

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

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, 6439710);