Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31);(1,2)(3,5)(4,6)(7,8)(10,11)(14,15)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32);(2,3)(6,7)(9,10,13,11)(12,14,16,15)(19,26)(22,25)(27,32)(30,31);(4,8)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32);(1,3,5,2)(4,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520462);