Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2076 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,18,13,19,12,33)(7,20,15,34,23,26,22,35)(8,24,27,36,14,29,32,21)(9,28,30,17,10,31,25,16);(1,2)(3,4)(5,7,11,22,13,23,12,15)(6,14,33,32,19,8,18,27)(9,29,30,36,10,24,25,21)(16,34,31,26,17,35,28,20);(2,4)(5,8,25,35,11,32,9,26,13,14,30,34,12,27,10,20)(6,15,28,21,33,23,16,24,19,22,31,36,18,7,17,29)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=h,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=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320894, #72320900, #72320963, … |
| maximal normal subgroup of | #72320894, #72320900, #72320963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56347);