#6366501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1499 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)(3,7)(4,10)(5,14)(6,9)(8,11)(12,15)(13,16)(17,18,24,30)(19,31,35,46)(20,26,21,27)(22,29,23,28)(25,39,42,34)(32,47,36,41)(33,43,48,40)(37,45,38,44);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,24,35)(18,25,30,42)(20,36,21,32)(22,34,23,39)(26,43,27,40)(28,31,29,46)(33,37,48,38)(41,44,47,45);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,24,21)(18,26,30,27)(19,32,35,36)(22,38,23,37)(25,40,42,43)(28,45,29,44)(31,47,46,41)(33,34,48,39);(2,9)(7,15)(8,16)(10,14)(17,21,24,20)(18,27,30,26)(19,33,35,48)(22,38,23,37)(25,41,42,47)(28,45,29,44)(31,43,46,40)(32,34,36,39);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(17,22,24,23)(18,28,30,29)(19,34,35,39)(20,37,21,38)(25,31,42,46)(26,44,27,45)(32,33,36,48)(40,41,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273697);