Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,9,14)(7,17,19,30,20,12,24,26)(8,13,23,29,21,27,22,28)(18,31,33,36,32,25,34,35);(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,25)(14,26)(15,17)(16,30)(21,32)(22,33)(23,34)(27,31)(28,35)(29,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(17,31)(19,33)(20,32)(24,34)(26,35)(30,36);(8,21)(13,27)(18,32)(22,23)(25,31)(28,29)(33,34)(35,36);(5,9,11,10)(6,14,15,16)(7,19,20,24)(8,22,21,23)(12,26,17,30)(13,28,27,29)(18,33,32,34)(25,35,31,36)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72262934, #72262943, #72262944, … |
| maximal normal subgroup of | #72262934, #72262943, #72262944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277899);