Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17,21,12)(8,22)(9,14)(10,25)(11,15)(13,23)(16,26)(18,28,19,32)(20,29,36,33)(24,30)(27,34,31,35);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,27)(14,28)(15,17)(16,29)(22,32)(23,34)(24,35)(25,33)(26,36)(30,31);(2,4)(5,8,11,9)(6,13,15,30)(7,18,21,19)(10,24,26,23)(12,27,17,31)(14,25,22,16)(20,35,36,34)(28,33,32,29);(5,9,11,8)(6,14,15,22)(7,19,21,18)(10,23,26,24)(12,28,17,32)(13,16,30,25)(20,34,36,35)(27,29,31,33)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287020);