Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,28)(26,27);(2,4)(13,15,17,25,18,26,16,24)(14,19,20,27,22,28,21,23);(5,6,7,8)(10,12);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,27,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72315658, #72315676, #72315682, … |
| maximal normal subgroup of | #72315658, #72315676, #72315682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261266);