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 | |
5033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(17,18)(19,28)(20,31)(21,32)(22,27)(23,25)(24,29)(26,30);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,27)(22,26)(23,29)(25,28)(31,32);(1,3,5,8)(2,6,7,4)(18,25)(19,29)(20,27)(26,32);(1,3)(2,4)(5,8)(6,7)(17,20,23,27)(18,22,25,31)(19,30,29,21)(24,26,28,32);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,23,30)(18,26,25,32)(19,27,29,20)(22,28,31,24);(17,22)(18,27)(19,26)(20,25)(21,24)(23,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7826882);