Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #357 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | C2×#1331 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,11,7,12,29,39,8,14,24,44,32,13,6,30)(3,15,18,48,20,50,31,49,16,47,19,56,21,57,53,38)(4,22,25,40,27,58,52,42,28,46,62,41,63,59,26,10)(9,33,35,55,37,61,36,17,34,23,43,54,45,60,64,51);(1,3)(2,9)(4,23)(5,19)(6,31)(7,20)(8,16)(10,38)(11,43)(12,37)(13,45)(14,34)(15,46)(17,52)(18,24)(21,32)(22,47)(25,55)(26,51)(27,60)(28,33)(29,53)(30,36)(35,44)(39,64)(40,48)(41,56)(42,49)(50,59)(54,62)(57,58)(61,63);(3,16)(9,34)(15,47)(17,51)(18,19)(20,21)(23,33)(31,53)(35,43)(36,64)(37,45)(38,49)(48,56)(50,57)(54,55)(60,61)Polycyclic presentation:
2^9|a^2=e,d^2=fhi,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,e^b=ei,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225304, #72225310, #72225311, … |
| maximal normal subgroup of | #72225304, #72225310, #72225311, … |
| maximal quotient of | #72226999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57265);