Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,10,7,11,9,8,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(1,3,4,2)(5,7)(6,10)(8,9)(11,12)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,27,21,26);(5,8,11,10)(6,7,9,12)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(5,9,8,7,11,6,10,12)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136729);