Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #380 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,7,9,8,10,11,12)(13,14)(15,24)(16,29,32,43)(17,33)(18,22)(19,23)(20,26)(21,31,36,30)(25,39,42,34)(27,35)(28,38)(37,41,44,40);(1,2)(3,4)(13,15,17,26,19,28,18,27)(14,20,22,24,23,35,33,38)(16,25,30,40,32,42,31,41)(21,34,29,37,36,39,43,44);(2,4)(13,16,17,30,19,32,18,31)(14,21,22,29,23,36,33,43)(15,25,26,40,28,42,27,41)(20,34,24,37,35,39,38,44)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27361);