Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2058 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,16,19,31)(8,21)(9,14)(10,25)(11,15)(12,18,27,35)(13,24)(17,32,20,30)(22,36)(23,29)(26,33,28,34);(5,7,8,18,10,17,24,33,11,19,22,35,9,20,23,34)(6,12,13,16,14,28,21,30,15,27,29,31,25,26,36,32);(1,2,3,4)(7,17,19,20)(8,22)(12,26,27,28)(13,29)(16,30,31,32)(18,34,35,33)(21,36)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237229, #72237285, #72238782, … |
| maximal normal subgroup of | #72237229, #72237285, #72238782, … |
| maximal quotient of | #72237231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56387);