Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#466 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,25)(9,15)(10,32)(11,34)(12,17)(13,22)(14,28)(16,19)(20,48)(21,43)(23,56)(24,45)(26,58)(27,40)(29,61)(30,39)(31,41)(33,62)(35,52)(36,54)(37,55)(38,47)(42,51)(44,46)(49,63)(50,57)(53,65)(59,60)(64,66)(67,68);(1,3,4,2)(5,7,19,46,11,23,33,55,12,24,31,54,9,21,10,22)(6,13,32,43,15,36,41,45,17,37,62,56,34,44,16,18)(8,20,47,66,29,52,28,51,30,53,60,68,26,49,27,50)(14,35,61,64,38,48,25,57,40,63,58,67,59,65,39,42);(5,8,10,27,9,26,31,60,12,30,33,28,11,29,19,47)(6,14,16,39,34,59,62,58,17,40,41,25,15,38,32,61)(7,20,22,50,21,49,54,68,24,53,55,51,23,52,46,66)(13,35,18,42,44,65,56,67,37,63,45,57,36,48,43,64);(5,9,12,11)(6,15,17,34)(7,21,24,23)(8,26,30,29)(10,31,33,19)(13,36,37,44)(14,38,40,59)(16,32,41,62)(18,43,45,56)(20,49,53,52)(22,54,55,46)(25,58,39,61)(27,60,28,47)(35,48,63,65)(42,64,57,67)(50,68,51,66)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402092);