Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C22 | #1326 |
| prime radical | C24 | #101 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(2,5,6,7)(9,10,13,19,15,20,35,49,16,21,36,50,14,37,34,65)(11,22,24,52,27,54,59,69,26,44,58,67,25,53,33,64)(12,28,30,61,31,18,43,45,32,46,63,70,38,47,66,71)(17,39,40,29,41,57,48,60,42,62,68,23,51,55,72,56);(1,2)(3,7)(4,6)(5,8)(9,11,15,25,16,26,14,27)(10,17,37,41,21,42,20,51)(12,29,38,60,32,23,31,56)(13,33,35,58,36,59,34,24)(18,44,46,54,47,22,28,53)(19,48,65,68,50,72,49,40)(30,62,66,55,63,39,43,57)(45,69,70,52,71,64,61,67)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fh,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1152);