Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4318 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)(5,6,7,8)(9,10,14,18)(11,15,17,13)(12,16,20,23)(19,22,24,21);(9,11,15,10,14,17,13,18)(12,19,22,16,20,24,21,23);(2,4)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23);(12,20)(16,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239933 |
| maximal normal subgroup of | #72239933 |
| maximal quotient of | #72239942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260594);