Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
285 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 285 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,13,17,16)(7,20,19,23)(8,14)(9,15)(10,26)(11,29)(18,35,36,34)(21,27)(22,28)(24,33,31,32)(25,30);(1,2,3,4)(5,6,14,33,30,36,27,23,29,7,21,35,26,31,8,16,12,17,15,32,10,18,28,20,11,19,22,34,25,24,9,13);(5,7,22,32,10,24,8,23,29,17,14,34,25,36,28,16,12,19,21,33,30,31,9,20,11,6,15,35,26,18,27,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eghi,d^2=ghi,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72216315, #72225566, #72225568, … |
| maximal normal subgroup of | #72216315, #72225566, #72225568, … |
| maximal quotient of | #72216305, #72216324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60180);