#9123872

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

5351 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)(13,16)(14,15)(17,18)(19,23)(20,24)(21,32)(22,26)(25,29)(27,28)(30,31);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,25,30,32);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,16)(13,14)(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,25,29,32);(3,8)(5,6)(10,16)(12,14)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,32,27,25);(1,2)(3,5)(4,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,25)(19,20)(22,29)(23,30)(24,27)(26,32)(28,31)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=gh,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, 9031068);