Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(13,14)(15,18)(16,19)(17,20)(21,22,25,27)(23,24,28,26);(6,10)(8,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,28)(25,27);(5,7,9,12)(6,11,10,8)(13,15,16,20)(14,18,19,17)(21,23,25,28)(22,26,27,24);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,20)(17,19)(21,23)(22,24)(25,28)(26,27);(5,8)(6,7)(9,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fh,c^2=f,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1587215);