#6351215

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

2648 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 6258411);