#6355102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1526 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:

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

Polycyclic presentation:

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

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, 6262298);