#6365331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365322
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

1686 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,28)(19,29,34,41)(20,25,37,45)(21,27,22,26)(24,32,42,33)(30,40,35,48)(31,39,47,43)(36,46,38,44);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,23,34)(18,24,28,42)(20,35,37,30)(21,33,22,32)(25,43,45,39)(26,29,27,41)(31,38,47,36)(40,46,48,44);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,30)(21,38)(22,36)(23,37)(24,39)(26,46)(27,44)(28,45)(29,40)(31,33)(32,47)(34,35)(41,48)(42,43);(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,40)(26,44)(27,46)(28,45)(29,39)(30,33)(32,35)(34,47)(41,43)(42,48);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,28,27)(19,32,34,33)(20,36,37,38)(24,41,42,29)(25,44,45,46)(30,31,35,47)(39,40,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,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,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272527);