Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,13)(12,16,22,24)(14,17)(15,18)(19,23,20,21);(1,2,3,5,4,6,7,8)(9,11,13,10,15,17,14,18)(12,20,21,24,22,19,23,16);(2,6)(5,8)(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,20)(18,24);(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,20,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108617);