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 | |
2405 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)(9,10,14,13)(11,16,12,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32);(6,8)(9,11,14,12)(10,15,13,16)(18,21)(20,29)(23,25)(26,30);(5,6)(7,8)(9,10,14,13)(11,15,12,16)(17,19)(18,23)(20,30)(21,25)(22,27)(24,32)(26,29)(28,31);(1,2,3,4)(9,12,14,11)(10,15,13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30);(9,13,14,10)(11,15,12,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,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, 1520888);