Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,19,38,59,42,63,13,21,9,17,12,20,44,64,14,22,40,61,37,58,39,60,24,47,10,18,25,62,66,68)(7,23,28,56,36,67,34,15,30,49,27,65,29,52,43,50,31,48,41,16,45,53,33,51,46,57,26,55,8,32,35,54);(1,2,3,4)(5,7,24,30,14,31,13,46)(6,15,47,50,22,51,21,54)(8,25,36,37,29,12,45,38)(9,34,11,43,10,33,40,35)(16,20,55,59,56,62,65,58)(17,52,19,53,18,32,61,67)(23,64,49,63,48,68,57,60)(26,44,28,42,27,66,41,39);(5,8)(6,16)(7,25)(9,28)(10,41)(11,27)(12,31)(13,45)(14,29)(15,20)(17,57)(18,49)(19,23)(21,65)(22,56)(24,36)(26,40)(30,37)(32,63)(33,39)(34,42)(35,44)(38,46)(43,66)(47,55)(48,61)(50,59)(51,62)(52,60)(53,64)(54,58)(67,68)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,f^c=fg,g^c=gh,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59721);