#9224239

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

5265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9131435);