Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28246 | 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 | |
361 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 361 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:
(2,3)(5,6)(7,18)(8,15)(9,14)(10,26)(11,27)(12,17)(13,20)(16,24)(19,34)(21,30)(22,31)(23,33)(25,35)(28,32)(29,36);(1,2)(3,4)(5,7)(6,13)(8,20)(9,25)(10,19)(11,22)(12,28)(14,30)(15,18)(16,34)(17,31)(21,23)(24,36)(26,29)(27,32)(33,35);(1,3,4,2)(6,14,15,33)(7,19,20,36)(9,10)(11,12)(16,27,26,17)(21,28,25,22)(23,24)(29,30)(31,32)(34,35)Polycyclic presentation:
2^9|c^2=dghi,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gi,c^b=cf,e^b=ei,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226860, #72226872, #72226873 |
| maximal normal subgroup of | #72226860, #72226872, #72226873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52353);