Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,14,17)(11,16,21,24)(12,19,13,18)(15,22,23,20);(5,6,8,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,18)(17,23)(19,24);(5,7,8,6)(9,11)(10,16)(12,22)(13,20)(14,21)(15,18)(17,24)(19,23);(1,3,4,2)(10,17)(15,23)(16,24)(18,19);(9,12,14,13)(10,18,17,19)(11,20,21,22)(15,16,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=i,b^a=bf,d^a=di,e^a=ei,f^a=fi,f^b=fi,e^c=ei164 conjugacy classes — view the full character table.
| maximal subgroup of | #72224708, #72226671, #72226673, … |
| maximal normal subgroup of | #72224708, #72226671, #72226673, … |
| maximal quotient of | C2×#6370083, #72240773, #72253211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277279);