Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129409 |
| 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,19,30)(8,20,23,13)(9,14)(10,15)(11,17)(12,18,29,24)(21,28)(22,34,36,26)(25,32)(27,33)(31,35);(1,2)(3,4)(5,7,10,19)(6,11,15,28)(8,21,23,17)(9,24,25,18)(12,26,29,34)(13,30,20,16)(14,31,32,27)(22,33,36,35);(2,4)(5,7)(6,12)(8,17)(9,24)(10,19)(11,26)(13,27)(14,30)(15,29)(16,32)(18,25)(20,31)(21,23)(22,35)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=ei,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267974, #72267978 |
| maximal normal subgroup of | #72267974, #72267978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54779);