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 | |
542 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)(7,16,19,29)(8,21)(9,25)(10,15)(11,18,28,35)(12,23)(13,31)(14,26)(17,20,34,32)(22,24)(27,33,36,30);(5,7,8,18,9,20,23,36,10,19,24,35,26,32,31,27)(6,11,12,16,14,30,21,17,15,28,13,29,25,33,22,34);(1,2,3,4)(5,6)(7,17,19,34)(8,22)(9,25)(10,15)(11,27,28,36)(12,31)(13,23)(14,26)(16,32,29,20)(18,30,35,33)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249883, #72249898, #72249926, … |
| maximal normal subgroup of | #72249883, #72249898, #72249926, … |
| maximal quotient of | #72265385, #72279865 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56440);