Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7349 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1317 |
| commutator, prime radical | C24 | C2×C16 |
| Frattini, cosocle | #2112 | C22 |
| socle | C2 | #7055 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2112 ◁ G | |
| LECS | G ▷ #2112 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 32 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 — 2 generators on 64 points:
(1,2,4,10,5,11,19,32,6,12,20,33,22,35,45,56,7,9,18,29,23,30,43,52,24,31,44,53,47,54,61,63)(3,8,14,25,15,27,37,48,16,28,38,49,39,51,58,62,13,26,17,40,21,34,41,55,36,50,42,59,46,57,60,64);(1,3)(2,8,9,26)(4,17)(5,21)(6,16)(7,13)(10,25,29,40)(11,34,30,27)(12,28,31,50)(14,18)(15,23)(19,37)(20,42)(22,46)(24,36)(32,55,52,48)(33,49,53,59)(35,57,54,51)(38,44)(39,47)(41,43)(45,58)(56,64,63,62)(60,61)Polycyclic presentation:
2^9|a^2=d,b^2=g,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ci,d^b=de,f^b=fi,d^c=dg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 882);