Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2338 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)(6,9)(7,11)(13,14,18,22,17,21,16,20)(15,19,26,28,25,23,24,27);(5,6,8,9)(7,10,11,12)(13,14,16,20,17,21,18,22)(15,23,24,28,25,19,26,27);(2,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,23)(22,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,22)(15,24,25,26)(19,27,23,28);(15,25)(19,23)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283010);