#6353205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2719 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260401);