Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(2,6,8,7)(11,19)(15,23)(18,24)(20,21);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225265, #72225275, #72225318 |
| maximal normal subgroup of | #72225265, #72225275, #72225318 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113559);