Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2396 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)(10,12)(13,14)(15,23)(16,25)(17,21)(18,22)(19,24)(20,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,20,22,23)(16,24,27,21)(17,28,19,25);(5,7,8,11)(6,10,9,12)(13,16,17,20,18,27,19,23)(14,15,24,28,22,26,21,25);(6,9)(10,12)(13,17,18,19)(14,21,22,24)(15,25,26,28)(16,20,27,23);(15,26)(16,27)(20,23)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fg,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287437);