Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,9,7)(8,11,12,10)(14,18)(15,19)(23,27)(26,28);(6,7)(8,12)(13,14)(15,16)(17,18)(19,20)(22,25)(26,28);(6,7)(10,11)(14,18)(15,19)(21,22,24,25)(23,26,27,28);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,16,18,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 492761);