Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(5,7,10,6)(8,12,9,11)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(5,8,10,9)(6,11,7,12)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(5,8,10,9)(6,12,7,11);(1,3,4,2)(5,9,10,8)(6,12,7,11)(14,21)(19,25)(20,28)(23,27)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,d^b=df,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6308091);