Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,28)(25,27);(1,2)(3,4)(6,9)(10,12);(2,4)(13,15,17,23)(14,18,19,27)(16,25,26,22)(20,28,24,21);(5,7,8,11)(6,10,9,12);(14,19)(18,27)(20,24)(21,28)Polycyclic presentation:
2^9|c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284934, #72285021, #72315234, … |
| maximal normal subgroup of | #72284934, #72285021, #72315234, … |
| maximal quotient of | #72315959, #72316046 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339224);