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 | |
454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,29,35,32,36,12,18,10,15,11,17,30,20)(7,19,21,8,24,25,27,33,23,28,14,26,22,34,31,13);(1,2)(3,4)(5,7,11,24,12,23,29,22)(6,13,17,8,18,33,35,26)(9,27,30,14,10,31,32,21)(15,34,36,19,16,25,20,28);(2,4)(5,8,11,33,12,26,29,13)(6,14,17,31,18,21,35,27)(7,20,24,15,23,36,22,16)(9,28,30,34,10,19,32,25)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=ghi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320894, #72320925, #72320963, … |
| maximal normal subgroup of | #72320894, #72320925, #72320963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56576);