#6353144

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

2824 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=h,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, 6260340);