Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,11)(9,12)(13,14,18,20,17,21,16,25)(15,22,24,19,23,28,26,27);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(15,23)(19,27)(22,28)(24,26);(2,4)(5,8,10,9)(6,11,7,12)(13,15)(14,19)(16,26)(17,23)(18,24)(20,28)(21,27)(22,25);(5,9,10,8)(6,11,7,12)(13,15)(14,19)(16,26)(17,23)(18,24)(20,28)(21,27)(22,25)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291040);