#9230906

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

4874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=i,f^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, 9138102);