Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1656 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,17,8,20)(9,14,24,31)(10,15)(11,16)(12,18)(13,22,30,19)(21,28)(23,29,36,26)(25,33)(27,34)(32,35);(1,2)(3,4)(5,7,11,8)(6,12,16,28)(9,18,24,21)(10,22,25,19)(13,29,30,26)(14,17,31,20)(15,32,33,27)(23,35,36,34);(2,4)(5,8,11,7)(6,13,16,30)(9,21,24,18)(10,19,25,22)(12,26,28,29)(14,27,31,32)(15,17,33,20)(23,34,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gh,e^c=ei,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267986, #72267997 |
| maximal normal subgroup of | #72267986, #72267997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54776);