Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,20)(11,21,12,17)(13,19,14,18)(16,23,24,22);(9,11,15,12)(10,16,20,24)(13,22,14,23)(17,18,21,19);(2,4);(5,7,8,6)(9,12,15,11)(10,17,20,21)(13,22,14,23)(16,18,24,19);(9,13,15,14)(10,18,20,19)(11,22,12,23)(16,21,24,17)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284172, #72284435, #72284589, … |
| maximal normal subgroup of | #72284172, #72284435, #72284589, … |
| maximal quotient of | #72305584, #72305617, #72305769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279871);