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 | |
478 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)(5,6,12,17,13,18,11,32)(7,19,22,26,23,25,21,8)(9,27,29,36,10,30,28,16)(14,15,33,20,24,35,34,31);(5,7,9,20,11,22,28,35,13,23,10,31,12,21,29,15)(6,14,16,19,17,34,27,8,18,24,36,25,32,33,30,26);(2,4)(5,8,9,24,11,25,28,33,13,26,10,14,12,19,29,34)(6,15,16,7,17,20,27,22,18,35,36,23,32,31,30,21)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250154, #72250167, #72250188, … |
| maximal normal subgroup of | #72250154, #72250167, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56085);