Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343485 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,15,17,21)(16,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(11,18)(16,23)(19,20)(22,24);(2,6)(5,7)(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72234285, #72258978, #72259001, … |
| maximal normal subgroup of | #72234285, #72258978, #72259001, … |
| maximal quotient of | #72233872, C2×#6343491 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250687);