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 | |
4791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(4,8)(6,7)(10,14)(15,16)(17,18)(19,21)(20,28)(22,26)(23,25)(24,31)(27,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,25)(18,23,26,21)(20,29,31,27)(24,30,28,32);(2,3)(6,7)(9,11,13,12)(10,15,14,16)(17,18,22,26)(19,21,25,23)(20,28,31,24)(27,32,29,30);(17,18)(19,23)(20,24)(21,25)(22,26)(27,32)(28,31)(29,30);(1,3,5,2)(4,7,8,6)(17,20,22,31)(18,24,26,28)(19,27,25,29)(21,30,23,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,25,26,19)(20,30,31,32)(24,29,28,27)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625297);