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 | |
4745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(20,28)(29,30);(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,29,32,30);(2,6)(3,7)(17,19,22,26)(18,21,23,24)(20,27,28,32)(25,30,31,29);(1,3)(2,4)(5,7)(6,8)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,32)(30,31);(1,4)(2,3)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,31)(24,29)(26,32);(17,21,22,24)(18,19,23,26)(20,29,28,30)(25,32,31,27)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,e^2=gi,f^2=i,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, 7849228);