Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | #2366 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,17,12,18,11,16)(7,19,23,32,24,36,22,13)(8,14,9,28,27,15,26,34)(20,33,25,31,29,35,21,30);(5,7)(6,13)(8,25)(9,29)(10,23)(11,22)(12,24)(14,33)(15,35)(16,36)(17,19)(18,32)(20,26)(21,27)(28,31)(30,34);(1,2,4,3)(5,8,11,26,12,27,10,9)(6,14,16,34,18,15,17,28)(7,20,23,25,24,29,22,21)(13,30,19,33,32,31,36,35);(1,3,4,2)(5,9,10,27,12,26,11,8)(6,15,16,28,18,14,17,34)(7,21,22,29,24,25,23,20)(13,31,19,35,32,30,36,33)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293994);