Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10)(11,19)(12,21)(13,16)(14,17)(15,18)(20,23)(22,24);(5,6,7,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,19)(17,24)(18,23);(1,3,4,2)(9,12)(10,11)(13,19)(14,24)(15,23)(16,20)(17,21)(18,22);(9,13,15,14)(10,16,18,17)(11,20,22,21)(12,19,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239936, #72240060 |
| maximal normal subgroup of | #72239936, #72240060 |
| maximal quotient of | #72239944, #72240049, #72240051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261122);