#6353250

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

2805 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,23,27)(19,28,31,39)(20,32,34,25)(21,26,22,36)(24,37,40,30)(29,44,47,43)(33,42,35,41)(38,46,48,45);(1,3)(2,6)(4,13)(5,11)(7,15)(8,16)(9,12)(10,14)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,47)(35,46)(41,43)(42,44);(4,14)(8,12)(9,16)(10,13);(2,7)(6,15)(8,12)(9,16)(17,20,23,34)(18,25,27,32)(19,29,31,47)(21,35,22,33)(24,38,40,48)(26,42,36,41)(28,43,39,44)(30,46,37,45);(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,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, 6260446);