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 | |
428 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)(5,6,11,18,12,20,30,19)(7,21,25,14,26,35,31,32)(8,27,24,13,23,33,22,34)(9,16,28,15,29,36,10,17);(5,7)(6,13,20,34)(8,9)(10,24)(11,31)(12,26)(14,15,32,17)(16,21,36,35)(18,27,19,33)(22,28)(23,29)(25,30);(5,8,28,31,30,22,9,26,12,23,10,25,11,24,29,7)(6,14,16,27,18,35,15,13,20,32,36,33,19,21,17,34)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fhi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250148, #72250158, #72250188, … |
| maximal normal subgroup of | #72250148, #72250158, #72250188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56345);