Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140477 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9622 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #471 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15)(4,21,26,56)(5,28)(6,11)(7,33)(8,14)(9,17)(10,25,41,60)(12,32)(13,31)(16,48,51,39)(18,47)(19,43)(20,35)(22,57,52,34)(23,45)(24,42,30,62)(27,40,61,44)(29,37)(36,53)(38,54)(46,50,63,49)(55,59,64,58);(1,3)(2,9)(4,22,26,52)(5,29)(6,17)(7,19)(8,20)(10,39,41,48)(11,43)(12,35)(13,37)(14,38)(15,28)(16,27,51,61)(18,31)(21,55,56,64)(23,54)(24,49,30,50)(25,59,60,58)(32,53)(33,36)(34,42,57,62)(40,63,44,46)(45,47);(1,4,23,60,7,27,5,30,8,26,6,25,31,61,32,24)(2,10,28,56,13,42,45,40,14,41,12,21,33,62,11,44)(3,16,17,50,19,22,53,59,20,51,54,49,18,52,29,58)(9,34,35,63,37,39,43,55,38,57,15,46,36,48,47,64)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=egh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,g^b=gi,d^c=di,f^c=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47673);