Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | #1726 | C22 |
| socle | C22 | #1290 |
| prime radical | C23 | #341 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ #1726 ▷ C22×C4 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,20,17,23,41,53,16,24,40,21,15,22,14,42)(11,25,27,54,31,56,28,64,30,57,38,67,29,55,63,72)(12,32,33,65,37,19,34,49,36,50,39,68,35,51,66,52)(18,43,44,61,47,70,71,60,48,62,45,59,46,58,69,26);(2,5,7,6)(9,11,14,28,15,29,40,27,16,30,41,63,17,31,13,38)(10,18,21,45,23,47,42,69,24,48,20,44,22,46,53,71)(12,26,34,58,35,59,33,62,36,60,66,70,37,61,39,43)(19,25,49,54,50,56,68,64,51,57,52,67,32,55,65,72)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=gh,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 551);