Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1196 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)(3,4)(5,6,8,7)(9,10,15,18,16,11,14,13)(12,17,22,20,23,19,21,24);(5,6,8,7)(9,11,16,10)(12,19,23,17)(13,14,18,15)(20,22,24,21);(2,4)(5,7,8,6)(9,12,15,22,16,23,14,21)(10,17,18,20,11,19,13,24);(9,13,14,11,16,18,15,10)(12,20,21,17,23,24,22,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,d^b=dei,e^b=ei,d^c=di,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264371);