Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143951 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1667 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,19)(8,13)(9,16)(10,15)(11,28)(12,18)(14,21)(17,25)(20,32)(22,35)(23,30)(24,31)(26,34)(27,33)(29,36);(1,2)(3,4)(5,7)(6,13)(8,10)(9,21)(11,23)(12,22)(14,17)(15,30)(16,19)(18,31)(20,25)(24,27)(26,36)(28,29)(32,33)(34,35);(2,4)(5,8,9,23,25,36,27,22,12,24,26,20,11,21,10,7)(6,14,15,32,28,31,34,35,18,29,33,30,17,13,16,19)Polycyclic presentation:
2^9|c^2=d,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,e^c=egh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52553);