Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(2,4)(9,10,11,12)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72302886, #72303128, #72303131 |
| maximal normal subgroup of | #72302886, #72303128, #72303131 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30627);