Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94556 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7349 |
| commutator | C23×C4 | C2×C8 |
| Frattini, cosocle | #1784 | C22 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1784 ◁ G | |
| LECS | G ▷ #1784 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 24 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:
(1,2,5,10,6,12,24,34,7,11,25,35,27,36,51,59)(3,13,15,39,17,40,45,62,16,41,44,61,46,55,63,64)(4,18,20,48,21,38,49,58,22,9,23,32,26,33,50,56)(8,28,29,47,30,52,54,14,31,42,53,43,37,60,57,19);(1,3)(2,8,11,31)(4,19)(5,15,25,44)(6,17)(7,16)(9,13,18,41)(10,29)(12,37,36,30)(14,22)(20,28,23,42)(21,47)(24,45,51,63)(26,43)(27,46)(32,39)(33,55,38,40)(34,57)(35,53)(48,61)(49,52,50,60)(54,59)(56,64)(58,62)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=i,d^2=f,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=chi,d^b=dei,e^b=ei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1755);