Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C42 |
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 | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2090 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)(5,6,7,8)(13,14)(15,21)(16,20)(17,22)(18,23)(19,24)(25,28)(26,27);(9,10,11,12)(13,15,18,16,19,26,17,25)(14,20,22,21,24,28,23,27);(2,4)(13,16,18,26,19,25,17,15)(14,21,22,28,24,27,23,20);(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72308677, #72308687, #72308727, … |
| maximal normal subgroup of | #72308677, #72308687, #72308727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260730);