Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,24,19,17,18,23)(15,25,16,27,26,21,28,22);(1,2,4,3)(6,9)(10,12)(13,15)(14,21)(16,20)(17,25)(18,28)(19,26)(22,24)(23,27);(1,3,4,2)(13,16)(14,22)(15,18)(17,27)(19,28)(20,26)(21,23)(24,25);(13,17,20,23,19,14,18,24)(15,25,16,27,26,21,28,22);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,17,24)(15,16,26,28)(21,22,25,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=f,g^2=i,b^a=bf,c^a=ci,c^b=ci,e^b=ei,e^c=ei200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277336);