Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(10,12)(14,19)(15,17)(20,23)(21,26)(22,24)(27,28);(1,3,4,2)(13,14)(15,19)(16,23)(17,26)(18,21)(20,24)(22,28)(25,27);(5,6,8,10)(7,12,11,9)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,27,28,23);(6,10)(9,12)(13,16,18,25)(14,20,21,28)(15,22,17,24)(19,23,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dhi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134249);