Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2033 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4)(5,6,12,20,13,21,36,51,14,22,11,19,10,18,9,17)(7,23,28,45,26,43,25,55,29,53,27,15,40,44,38,42)(8,30,34,48,33,50,32,54,35,52,39,65,37,16,46,49)(24,57,31,63,60,56,64,41,61,47,62,66,59,68,58,67);(5,7,10,26,14,29,13,40)(6,15,18,42,22,45,21,55)(8,31,33,58,35,62,37,64)(9,27,11,38,36,28,12,25)(16,47,30,56,50,57,52,68)(17,43,19,53,51,44,20,23)(24,46,59,34,61,32,60,39)(41,65,63,49,67,48,66,54);(5,8,9,32,10,37,11,34,14,35,36,46,13,33,12,39)(6,16,17,48,18,52,19,49,22,50,51,65,21,30,20,54)(7,24,25,58,40,59,28,62,29,61,38,64,26,60,27,31)(15,41,23,56,55,63,44,57,45,67,53,68,42,66,43,47)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56237);