#6351192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2602 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^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,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258388);