#9161121

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

5152 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068317);