#6571691

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478887);