Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2389 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)(6,10)(11,12)(13,14,19,23,18,22,24,25)(15,17,21,28,26,16,20,27);(5,6,9,10)(7,11,8,12)(13,15,24,20,18,26,19,21)(14,16,25,28,22,17,23,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,20,22,21)(15,25,26,23)(19,27,24,28);(5,8,9,7)(6,11,10,12)(13,17,18,16)(14,21,22,20)(15,23,26,25)(19,28,24,27);(14,22)(16,17)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=dfi,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289361);