Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| 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 | |
1209 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)(11,22)(12,23)(13,18)(14,20)(15,19)(16,21)(17,24);(5,6,8,7)(9,11)(10,13)(12,21)(14,23)(15,18)(16,17)(19,24)(20,22);(1,2,4,3)(5,7,8,6)(9,12)(10,17)(11,21)(13,16)(14,24)(15,22)(18,20)(19,23);(1,3,4,2)(9,13,16,12)(10,18,21,23)(11,20,17,19)(14,24,15,22)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,d^c=di,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264323);