Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
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 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,22)(19,21)(23,24);(10,14,17,16)(15,21,22,23);(1,2,3,4)(11,18)(15,22)(19,24)(21,23);(9,11,12,19,13,18,20,24)(10,15,14,21,17,22,16,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=eh,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72213041, #72216155, #72216157, … |
| maximal normal subgroup of | #72213041, #72216155, #72216157, … |
| maximal quotient of | #72213009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262172);