Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,13,17)(11,19)(12,15)(14,18,22,24)(16,20)(21,23);(1,3,6,2)(4,8,5,7)(9,11,13,12)(10,15)(14,21,22,20)(16,18)(17,19)(23,24);(9,12,13,11)(10,16,17,23)(14,20,22,21)(15,18,19,24);(1,4,6,5)(2,7,3,8)(10,17)(15,19)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=h,d^2=hi,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104457);