#8199216

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8106412);