#6357938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357891
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265134);