Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,3,4)(6,11)(8,20)(9,23)(12,25)(13,21)(14,27)(15,26)(16,31)(17,32)(19,33)(22,34)(24,35)(28,29)(30,36);(5,6)(7,16,18,30)(8,21,24,27)(9,14)(10,15)(11,20,26,35)(12,19,29,33)(13,23)(17,25)(22,28)(31,34)(32,36);(5,7)(6,12,15,29)(8,22)(9,19)(10,18)(11,25,26,28)(13,30,14,16)(17,24)(20,34)(21,36,27,31)(23,33)(32,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,g^b=gi,d^c=di,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240808, #72250343, #72250365, … |
| maximal normal subgroup of | #72240808, #72250343, #72250365, … |
| maximal quotient of | #72240813, #72264541 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34809);