Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×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×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18,22,26,23,27,21,31)(19,28,29,32,30,25,20,24);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,16,15,14);(1,4,6,5)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,19,21,20,23,30,22,29)(18,24,26,28,27,32,31,25);(1,5,6,4)(2,7,8,3)(10,13)(14,16)(17,20,21,30,23,29,22,19)(18,25,26,24,27,28,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=efg,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294567);