Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2578 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)(7,15,18,11)(8,19)(9,22)(10,13)(12,21)(14,24)(16,30,31,33)(17,26,23,29)(20,34)(25,32,27,36)(28,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,23)(10,18)(12,27)(13,15)(14,29)(17,24)(19,33)(20,31)(21,36)(22,26)(25,28)(30,34)(32,35);(2,4)(5,8)(6,12,13,28)(7,16)(9,21)(10,20)(11,25,15,27)(14,19,22,34)(17,32)(18,31)(23,36)(24,35)(26,33,29,30);(8,20)(12,28)(16,31)(19,34)(21,35)(25,27)(30,33)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265077);