Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(2,4)(5,6,9,12,10,16,26,28,11,17,8,22,23,15,24,14)(7,18,20,33,19,27,35,31,21,34,25,13,29,30,36,32);(1,2,3,4)(6,12,15,14,17,22,16,28)(7,19,21,29)(8,9)(13,18,32,27,33,34,31,30)(20,36,25,35)(24,26);(5,7,8,20,10,19,24,35,11,21,9,25,23,29,26,36)(6,13,14,18,15,31,22,30,17,33,28,34,16,32,12,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224944, #72224945, #72224954, … |
| maximal normal subgroup of | #72224944, #72224945, #72224954, … |
| maximal quotient of | #72225174, #72225222, #72227212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47949);