Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
718 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)(7,20)(8,26,29,45)(9,17)(10,16)(11,15)(12,19)(13,22)(14,28,41,31)(18,32)(21,49,52,67)(23,47)(24,36)(25,48)(27,59,61,40)(30,42,60,65)(33,43)(34,44)(35,62,63,53)(37,55)(38,57)(39,56)(46,51,66,54)(50,58,68,64);(2,4)(5,7,11,24,10,23,33,56,12,25,34,57,32,55,9,22)(6,13,17,37,18,38,44,48,19,39,43,47,16,36,15,20)(8,21,30,53,31,54,61,68,29,52,60,62,28,51,27,50)(14,35,42,49,26,58,59,66,41,63,65,67,45,64,40,46);(5,8,10,31,12,29,32,28)(6,14,16,45,19,41,18,26)(7,21,23,54,25,52,55,51)(9,27,11,30,33,61,34,60)(13,35,36,64,39,63,38,58)(15,40,43,65,44,59,17,42)(20,46,47,67,48,66,37,49)(22,50,24,53,56,68,57,62)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,e^c=ei,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54608);