Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(9,10,11,12)(14,18)(15,22)(16,23)(19,24)(20,28)(21,27)(25,26);(5,6)(7,8)(13,14,19,28,16,20,15,18,17,21,22,26,23,25,24,27);(1,2,3,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=dhi,c^2=e,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322385, #72322616, #72322632, … |
| maximal normal subgroup of | #72322385, #72322616, #72322632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56399);