Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8350 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #525 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,5,10,6,12,26,40,7,11,27,39,20,34,49,61)(3,13,16,44,17,33,24,55,15,43,48,58,28,51,56,64)(4,18,22,38,23,42,54,63,21,9,25,35,45,36,57,62)(8,29,30,14,32,46,60,19,31,52,59,47,41,50,37,53);(1,3,7,15)(2,8,11,31)(4,19,21,53)(5,24,27,56)(6,28,20,17)(9,33,18,51)(10,37,39,60)(12,41,34,32)(13,42,43,36)(14,45,47,23)(16,49,48,26)(22,50,25,46)(29,54,52,57)(30,40,59,61)(35,44,38,58)(55,62,64,63);(9,18)(13,43)(33,51)(35,38)(36,42)(44,58)(55,64)(62,63)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281586, #72281615, #72281659, … |
| maximal normal subgroup of | #72281586, #72281615, #72281659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37482);