#6476516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470355
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(17,18,22,26)(19,27,31,32)(20,24,21,25)(23,30,28,29);(9,11)(10,13)(12,15)(14,16)(17,19,21,29,22,31,20,30)(18,23,24,27,26,28,25,32);(2,5)(6,8)(11,15)(13,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,31)(27,30)(29,32);(11,15)(13,16)(17,20,22,21)(18,24,26,25)(19,29,31,30)(23,32,28,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,21)(18,25,26,24)(19,30,31,29)(23,32,28,27)

Polycyclic presentation:

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

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, 6383712);