#6352405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352341
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

2980 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:

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

Polycyclic presentation:

2^9|a^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,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, 6259601);