Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(10,11)(15,19)(17,20);(5,6,8,9)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,22)(23,25)(24,27)(26,28);(7,12)(10,11)(13,15,16,19)(14,17,18,20)(22,25)(23,28)(26,27);(5,7,8,12)(6,10,9,11)(14,18)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298582);