Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140362 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1941 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11220 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,13,7,14,29,10)(3,15,17,8,20,37,19,38)(4,21,23,45,26,59,27,47)(6,30,28,11,33,43,61,12)(9,32,41,24,42,50,40,25)(16,49,51,64,54,63,53,39)(18,35,55,44,57,36,31,46)(22,48,60,58,52,62,56,34);(1,3,5,17,7,20,29,19)(2,8,10,15,14,38,13,37)(4,22,23,60,26,52,27,56)(6,31,28,18,33,55,61,57)(9,39,40,63,42,64,41,49)(11,44,30,35,12,46,43,36)(16,50,51,25,54,32,53,24)(21,58,47,48,59,34,45,62);(1,4,7,26)(2,9,14,42)(3,16)(5,27,29,23)(6,32,33,50)(8,34)(10,41,13,40)(11,45,12,47)(15,48)(17,53)(18,56)(19,51)(20,54)(21,30,59,43)(22,31)(24,28,25,61)(35,63)(36,49)(37,62)(38,58)(39,46)(44,64)(52,55)(57,60)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56000);