Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2460 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,21,17,20,26,28)(15,22,24,25,23,19,27,16);(6,9)(7,11);(2,4)(13,15)(14,19)(16,21)(17,23)(18,24)(20,22)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(21,28)(24,27);(6,9)(10,12)(13,16,18,22,17,25,26,19)(14,15,21,24,20,23,28,27)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72284417, #72284482, #72284746 |
| maximal normal subgroup of | #72284417, #72284482, #72284746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306862);