#6351056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

2583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258252);