Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632036 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,23)(20,25,21,24)(28,31,30,32);(1,2,4,5)(3,6,7,8)(10,13)(14,15)(17,19,22,29)(18,23,26,27)(20,30,21,28)(24,32,25,31);(2,5)(6,8)(10,14)(12,16)(13,15);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32);(17,21,22,20)(18,25,26,24)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72282613, #72282855, #72283031 |
| maximal normal subgroup of | #72282613, #72282855, #72283031 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6565713);