Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #34743 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2293 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)(6,9)(7,11)(13,14,18,22,17,15,19,23)(16,21,25,28,24,20,26,27);(5,6,8,9)(7,10,11,12)(13,15,18,23,17,14,19,22)(16,20,25,27,24,21,26,28);(2,4)(6,9)(10,12)(13,16)(14,20)(15,21)(17,24)(18,25)(19,26)(22,27)(23,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,20)(17,24)(18,25)(19,26)(22,28)(23,27);(16,24)(20,21)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283105);