#6351155

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

2708 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)(19,30,34,40)(20,26,37,42)(21,28,22,27)(24,32,41,33)(25,36,43,35)(31,45,48,38)(39,47,44,46);(1,2,6,9)(3,8,10,15)(4,7,5,13)(11,14,12,16)(17,19,23,34)(18,24,29,41)(20,31,37,48)(21,32,22,33)(25,38,43,45)(26,39,42,44)(27,40,28,30)(35,46,36,47);(1,3,6,10)(2,8,9,15)(4,11,5,12)(7,14,13,16)(17,20,23,37)(18,25,29,43)(19,31,34,48)(21,35,22,36)(24,38,41,45)(26,28,42,27)(30,44,40,39)(32,46,33,47);(3,10)(8,15)(11,12)(14,16)(17,20)(18,26)(19,31)(21,36)(22,35)(23,37)(24,39)(25,27)(28,43)(29,42)(30,45)(32,47)(33,46)(34,48)(38,40)(41,44);(1,4,6,5)(2,7,9,13)(3,11,10,12)(8,14,15,16)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,35,37,36)(24,40,41,30)(25,26,43,42)(31,46,48,47)(38,39,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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, 6258351);