#6579817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,25,31)(20,27,28,24)(26,32,30,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(19,21)(20,28)(25,31)(26,32)(29,30);(10,14)(13,16)(17,19)(18,21)(20,29)(22,25)(23,31)(24,30)(26,27)(28,32);(1,3,4,7)(2,6,5,8)(17,20,22,28)(18,24,23,27)(19,26,25,30)(21,29,31,32);(17,21,22,31)(18,25,23,19)(20,30,28,26)(24,32,27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487013);