#6586817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(11,14)(12,13)(15,16)(17,18,22,23)(19,26,28,24)(20,21,25,31)(27,30,32,29);(2,5)(6,8)(17,19,22,28)(18,24,23,26)(20,30,25,29)(21,27,31,32);(10,13)(14,16)(17,20)(18,21)(19,27)(22,25)(23,31)(24,29)(26,30)(28,32);(9,11,12,15)(10,14,13,16)(18,23)(19,28)(20,31)(21,25)(27,30)(29,32);(1,3,4,7)(2,6,5,8)(17,21,22,31)(18,25,23,20)(19,29,28,30)(24,32,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6494013);