Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,19)(12,20)(13,16)(14,23)(15,18)(17,24)(21,22);(5,6,8,7)(9,11,15,21)(10,13,18,12)(14,20,24,16)(17,22,23,19);(1,2,4,3)(5,7,8,6)(9,12)(10,11)(13,15)(14,22)(16,17)(18,21)(19,24)(20,23);(1,3,4,2)(9,13)(10,16)(11,17)(12,15)(14,22)(18,20)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=ghi,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,e^b=ei,f^c=fi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264321);