Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14341 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,8)(6,11)(7,14)(10,18)(13,25)(15,21)(16,22)(17,24)(19,28)(20,26)(23,29)(27,31)(30,32);(1,3,5,15,6,14,18,28)(2,7,10,21,11,12,8,19)(4,13,16,27,17,26,29,32)(9,20,23,31,24,25,22,30);(2,8,11,10)(7,19,12,21)(9,22,24,23)(13,26)(20,31,25,30)(27,32);(1,4)(2,9)(3,13)(5,16)(6,17)(7,20)(8,22)(10,23)(11,24)(12,25)(14,26)(15,27)(18,29)(19,30)(21,31)(28,32)Polycyclic presentation:
2^9|b^2=f,c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338478, #72338484, #72338500 |
| maximal normal subgroup of | #72338478, #72338484, #72338500 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289909);