Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11)(8,19,21,12)(9,14)(10,13)(15,33,34,25)(16,35,36,26)(17,28)(18,27)(20,32,23,29)(22,31)(24,30);(1,2)(3,4)(5,7,10,18)(6,11,13,27)(8,16,21,36)(9,22,24,17)(12,26,19,35)(14,31,30,28)(15,23,34,20)(25,32,33,29);(5,8)(6,12)(7,15)(9,23)(10,21)(11,25)(13,19)(14,32)(16,22)(17,36)(18,34)(20,24)(26,31)(27,33)(28,35)(29,30);(1,3,4,2)(5,8)(6,12)(7,16)(9,20)(10,21)(11,26)(13,19)(14,29)(15,22)(17,34)(18,36)(23,24)(25,31)(27,35)(28,33)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290681);