#6350370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6257566);