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 |
| prime radical | #36019 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(17,18)(19,24)(20,26)(21,30)(22,27)(23,25)(28,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,25,31)(20,27,30,32)(21,29,26,22);(1,3,5,8)(2,6,7,4)(18,25)(19,28)(20,30)(22,29);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,23,30)(18,26,25,21)(19,27,28,32)(22,31,29,24);(1,4)(2,3)(5,6)(7,8)(17,21,23,26)(18,20,25,30)(19,29,28,22)(24,27,31,32);(17,22,23,29)(18,27,25,32)(19,26,28,21)(20,31,30,24)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756882);