Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137558 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9061 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| 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 | |
435 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,14,8,13,32,12,9,16,31,46,7,15,5,30)(3,17,21,10,20,36,19,38,23,48,55,39,22,44,33,37)(4,24,26,57,28,43,25,45,29,42,58,40,27,11,34,41)(18,47,51,60,50,59,64,61,53,63,54,56,52,62,49,35);(1,3,7,20,9,23,8,22)(2,10,13,37,16,39,15,38)(4,18,28,52,29,53,27,50)(5,21,31,19,32,55,6,33)(11,35,42,60,43,61,24,56)(12,44,46,48,30,36,14,17)(25,54,58,64,34,51,26,49)(40,47,45,59,57,63,41,62);(1,4)(2,11)(3,18)(5,26)(6,34)(7,28)(8,27)(9,29)(10,35)(12,45)(13,42)(14,40)(15,24)(16,43)(17,47)(19,54)(20,52)(21,49)(22,50)(23,53)(25,31)(30,41)(32,58)(33,51)(36,62)(37,60)(38,56)(39,61)(44,59)(46,57)(48,63)(55,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283166, #72283214, #72283264, … |
| maximal normal subgroup of | #72283166, #72283214, #72283264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44755);