Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52169 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1873 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)(6,9)(7,11)(13,14,19,23,18,17,22,28)(15,20,16,25,24,27,26,21);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,27)(18,24)(21,28)(22,26)(23,25);(2,4)(6,9)(10,12)(13,16)(14,21)(15,22)(17,25)(18,26)(19,24)(20,23)(27,28);(5,7,8,11)(6,10,9,12)(15,24)(16,26)(20,27)(21,25);(13,17,19,28,18,14,22,23)(15,20,16,25,24,27,26,21)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,d^b=di,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285403);