#8716834

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

5107 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8624030);