#6353089

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×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2698 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)(4,5)(8,15)(9,12)(10,13)(14,16)(17,18)(19,28)(20,25)(21,33)(22,36)(23,27)(24,29)(26,35)(30,45)(31,38)(32,42)(34,41)(37,40)(39,46)(43,47)(44,48);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15)(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);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15)(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);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,39)(25,41)(27,33)(28,43)(29,44)(31,47)(36,42)(37,46)(38,45)(40,48);(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|c^2=h,d^2=i,e^2=hi,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, 6260285);