Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,29,19,30,25,12,17,9,28,11,16,31,35)(7,18,20,34,32,8,24,13,21,33,22,36,23,27,14,26);(1,2)(3,4)(5,7,11,32,12,21,29,23)(6,13,16,36,17,26,19,34)(8,25,33,15,27,35,18,28)(9,14,30,20,10,24,31,22);(2,4)(5,8,9,26,29,33,31,34,12,27,10,13,11,18,30,36)(6,14,15,7,19,20,25,32,17,24,28,21,16,22,35,23)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321762, #72321789, #72321804, … |
| maximal normal subgroup of | #72321762, #72321789, #72321804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56288);