#6354769

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354705
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

1617 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,35)(30,44)(31,40)(32,39)(36,41)(37,42)(38,45)(43,47)(46,48);(1,3,5,13,6,12,4,11)(2,7,8,10,9,15,14,16)(17,19,21,33,23,31,22,32)(18,24,27,29,28,39,26,40)(20,30,35,47,36,45,37,46)(25,38,34,43,42,44,41,48);(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)(39,44)(40,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(3,12)(7,15)(10,16)(11,13)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,29,39,40)(25,41,42,34)(30,46,45,47)(38,43,44,48)

Polycyclic presentation:

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

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, 6261965);