#6351910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2734 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259106);