Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2076 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,10,16,11,17,36,53,12,18,37,54,20,40,9,15)(7,19,24,45,25,51,22,55,26,42,34,13,38,43,23,44)(8,27,31,50,32,52,30,56,33,47,35,14,39,48,29,49)(21,58,60,67,61,68,59,46,62,41,63,57,64,66,28,65);(1,2)(3,4)(5,7,20,25,12,26,11,38)(6,13,40,44,18,45,17,55)(8,28,32,63,33,59,39,60)(9,34,37,23,36,24,10,22)(14,46,49,67,50,65,56,57)(15,51,54,42,53,43,16,19)(21,35,64,29,62,31,61,30)(27,41,52,68,47,58,48,66);(2,4)(5,8,20,32,12,33,11,39)(6,14,40,49,18,50,17,56)(7,21,25,64,26,62,38,61)(9,35,37,29,36,31,10,30)(13,41,44,68,45,58,55,66)(15,52,54,47,53,48,16,27)(19,57,51,46,42,67,43,65)(22,60,34,28,23,63,24,59)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56573);