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 | |
5104 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,26)(21,25)(22,24)(27,30)(28,32)(29,31);(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32);(2,6)(3,7)(9,12,13,10)(11,15,16,14)(18,24)(19,26)(21,29)(28,30);(1,3,5,7)(2,4,6,8)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,31)(30,32);(1,4)(2,7)(3,6)(5,8)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7662040);