Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1365 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7121 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,12,8,15,29,51,6,16,4,24,27,14,30,50,9,17,28,11,26,45,57,47,35,13,31,48,7,36,33,46)(3,18,19,52,22,54,59,43,20,55,58,53,21,49,25,39,23,56,60,40,37,64,32,10,34,38,62,41,61,44,63,42);(1,3,9,23)(2,10)(4,25,31,63)(5,32,28,59)(6,34,35,20)(7,37,27,22)(8,21,26,61)(11,44)(12,49)(13,42)(14,41)(15,52)(16,39)(17,43)(18,46)(19,57,60,29)(24,54)(30,62,33,58)(36,53)(38,51)(40,45)(47,55)(48,64)(50,56)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=def,e^b=ei,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1135);