Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,12,14)(11,13,16,15)(18,22)(19,25)(20,28)(30,32);(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,26,25,23)(20,29,28,24)(27,30,31,32);(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2,3,4)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(9,11,12,16)(10,13,14,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fg,d^2=fi,e^2=f,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, 1520887);