#6358717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358586
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

1541 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,23,28)(19,29,30,43)(20,25,35,42)(21,27,22,26)(24,36,38,32)(31,45,46,40)(33,48,47,39)(34,44,37,41);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,30)(18,24,28,38)(20,33,35,47)(21,32,22,36)(25,40,42,45)(26,29,27,43)(31,37,46,34)(39,44,48,41);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(19,30)(24,38)(29,43)(31,46)(32,36)(33,47)(39,48)(40,45);(17,20)(18,25)(19,31)(21,34)(22,37)(23,35)(24,39)(26,41)(27,44)(28,42)(29,45)(30,46)(32,47)(33,36)(38,48)(40,43);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,21,23,22)(18,26,28,27)(19,32,30,36)(20,34,35,37)(24,29,38,43)(25,41,42,44)(31,47,46,33)(39,45,48,40)

Polycyclic presentation:

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

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, 6265913);