Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2065 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,17,10,14,25,13,11,16,26,15,8,23,24,32)(7,18,21,28,22,31,36,29,19,34,35,33,20,12,27,30);(5,7,8,20,11,19,10,22)(6,12,14,18,16,31,23,34)(9,21,24,27,26,35,25,36)(13,28,15,29,32,33,17,30);(2,4)(6,13,16,32)(7,19)(8,10)(9,25)(12,28,31,33)(14,17,23,15)(18,30,34,29)(21,27)(24,26)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250150, #72250163, #72250188, … |
| maximal normal subgroup of | #72250150, #72250163, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56068);