Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #165149 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36123 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,21,22,18)(16,19)(17,20)(23,27,28,25)(24,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(15,22)(18,21)(23,28)(25,27);(6,11)(7,8)(10,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27);(2,4)(6,12)(10,11)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128983);