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 | |
5240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18)(19,24)(20,23)(21,26)(22,25)(27,31)(28,30)(29,32);(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(18,20)(19,26)(22,30)(29,31);(1,2)(3,6)(4,5)(7,8)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31);(1,3)(2,6)(4,8)(5,7)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31);(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7655905);