Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94419 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7304 |
| commutator | C22×Q8 | C42 |
| Frattini, cosocle | C2×#536 | C22 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#536 ◁ G | |
| LECS | G ▷ C2×#536 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(1,2,5,10)(3,13,15,43)(4,18,20,53)(6,27,29,39)(7,11,25,40)(8,31,32,48)(9,35,36,21)(12,26,42,30)(14,46,47,38)(16,45,51,64)(17,44,49,59)(19,34,54,62)(22,41,24,57)(23,56,52,61)(28,37,55,63)(33,58,60,50);(1,3)(2,8,11,33)(4,19)(5,23,25,51)(6,16)(7,17)(9,13,37,44)(10,38)(12,34,27,46)(14,22)(15,26,49,29)(18,45,41,61)(20,31,24,58)(21,50)(28,48)(30,52)(32,42)(35,54,55,47)(36,56)(39,60)(40,62)(43,53)(57,59)(63,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=df,h^b=hi,d^c=dhi,f^c=fi,e^d=ei,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72306864, #72306865, #72306871, … |
| maximal normal subgroup of | #72306864, #72306865, #72306871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1615);