Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,14,28,13)(7,16,18,9,19,27,32,26)(8,20,23,31,25,30,24,12)(17,22,34,36,21,29,35,33);(1,2)(3,4)(5,7,10,18,11,19,28,32)(6,9,13,16,14,26,15,27)(8,21,23,35,25,17,24,34)(12,29,30,36,31,22,20,33);(2,4)(5,8,10,23,11,25,28,24)(6,12,15,20,14,31,13,30)(7,17,18,34,19,21,32,35)(9,22,27,36,26,29,16,33);(5,9)(6,7)(8,22)(10,27)(11,26)(12,17)(13,32)(14,19)(15,18)(16,28)(20,34)(21,31)(23,36)(24,33)(25,29)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295471);