Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28502 | 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 | |
517 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(5,6,11,16)(7,17,20,34)(8,22,26,31)(9,15,10,14)(12,27,29,19)(13,23,33,25)(18,32,21,35)(24,28,36,30);(1,2)(3,4)(7,18)(8,23)(9,10)(12,28)(13,31)(14,15)(17,34)(19,27)(20,21)(22,33)(25,26)(29,30);(2,4)(5,7)(6,12)(8,24)(9,21)(10,18)(11,20)(13,32)(14,30)(15,28)(16,29)(17,22)(19,25)(23,27)(26,36)(31,34)(33,35)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283166 |
| maximal normal subgroup of | #72283166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50400);