Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| 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 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12,15,26)(7,17,19,23)(8,21)(9,22)(10,24)(13,28)(14,31,27,16)(18,33,34,20)(25,32)(29,35);(1,2)(3,4)(5,6,8,12,10,16,21,27,11,15,22,26,24,31,9,14)(7,13,18,25,20,30,17,29,19,28,34,35,33,36,23,32);(2,4)(5,7,8,18,10,20,21,17,11,19,22,34,24,33,9,23)(6,13,12,25,16,30,27,29,15,28,26,35,31,36,14,32)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239804, #72239809, #72239817, … |
| maximal normal subgroup of | #72239804, #72239809, #72239817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56075);