Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200418 |
| 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 | C2×C42 | C24 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,14,17,15,18,13,11)(12,20,22,21,23,16,24,19);(1,2,6,8,4,7,3,5)(9,11,14,18,15,17,13,10)(12,21,24,16,23,19,22,20);(2,7)(5,8)(9,12)(10,16)(11,19)(13,22)(14,24)(15,23)(17,21)(18,20);(9,13,15,14)(10,17,18,11)(12,22,23,24)(16,21,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338724, #72338835 |
| maximal normal subgroup of | #72338724, #72338835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107774);