Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(9,10,11,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72313239, #72313253, #72313259, … |
| maximal normal subgroup of | #72313239, #72313253, #72313259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265096);