Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132501 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,6,12,33,44,5,26)(3,15,19,9,18,36,17,51)(4,22,23,38,57,40,59,42)(7,14,29,11,8,13,30,43)(10,32,39,25,56,28,58,24)(16,47,48,63,53,64,60,55)(20,41,31,35,21,46,27,37)(34,50,61,54,62,49,45,52);(1,3,8,21)(2,9,13,37)(4,16,24,50)(5,27,29,19)(6,31,30,17)(7,20,33,18)(10,34,22,55)(11,41,26,36)(12,15,43,46)(14,35,44,51)(23,52,32,60)(25,49,57,53)(28,48,59,54)(38,64,39,45)(40,63,56,62)(42,47,58,61);(1,4,8,24)(2,10)(3,16,21,50)(5,28,29,59)(6,32,30,23)(7,25,33,57)(9,34)(11,42)(12,39)(13,22)(14,40)(15,45)(17,52,31,60)(18,53,20,49)(19,54,27,48)(26,58)(35,63)(36,61)(37,55)(38,43)(41,47)(44,56)(46,64)(51,62)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40204);