Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,21)(16,25)(17,19)(18,24)(20,26)(22,27)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15,16,24,17,23,26,27)(14,18,20,21,19,22,25,28);(2,4)(6,9)(10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,27)(24,28)(25,26);(5,7,8,11)(6,10,9,12)(14,19)(18,22)(20,25)(21,28);(15,23)(18,22)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342743);