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 | #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 | |
1148 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)(5,6,11,16,12,18,10,17)(7,19,22,13,23,30,28,31)(8,14,25,33,26,15,9,27)(20,34,24,29,35,32,21,36);(5,7)(6,13)(8,24)(9,20)(10,28)(11,22)(12,23)(14,32)(15,34)(16,30)(17,19)(18,31)(21,26)(25,35)(27,29)(33,36);(1,2,4,3)(5,8,10,9,12,26,11,25)(6,14,17,27,18,15,16,33)(7,20,22,24,23,35,28,21)(13,29,30,32,31,36,19,34);(1,3,4,2)(5,9,10,26,12,25,11,8)(6,15,16,27,18,14,17,33)(7,21,22,20,23,24,28,35)(13,29,19,34,31,36,30,32)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=gi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293995);