Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201424 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1577 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)(3,4)(5,8)(6,7)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(1,3,4,2,6,8,5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,4,6,5)(2,3,7,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257416, #72257500, #72257545, … |
| maximal normal subgroup of | #72257416, #72257500, #72257545, … |
| maximal quotient of | #72264137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108626);