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)(11,16)(12,18)(13,17)(14,19)(15,20)(21,24)(22,23);(9,11)(10,15)(12,20)(13,22)(14,21)(16,17)(18,24)(19,23);(2,4);(5,7,8,6)(9,11)(10,16)(12,22)(13,20)(14,21)(15,17)(18,23)(19,24);(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,16,23,24)Polycyclic presentation:
2^9|a^2=h,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, #72284699 |
| maximal normal subgroup of | #72284172, #72284699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279870);