Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143196 |
| 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 | C2×#135 | D4 |
| 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 | |
505 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 505 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:
(5,6,11,7)(8,18,21,34)(9,23,28,24)(10,17,26,14)(12,29,15,20)(13,32,16,27)(19,33,22,35)(25,30,36,31);(1,2)(3,4)(5,7,11,6)(8,15,21,12)(9,24,28,23)(10,14,26,17)(13,27,16,32)(18,25,34,36)(19,30,22,31)(20,35,29,33);(2,4)(5,8)(6,12)(7,15)(9,25)(10,19)(11,21)(13,33)(14,30)(16,35)(17,31)(18,23)(20,27)(22,26)(24,34)(28,36)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50395);