Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,19)(12,16)(13,17)(14,23)(15,18)(20,22)(21,24);(2,5)(6,7)(9,11,14,16,15,21,13,20)(10,12,17,19,18,22,23,24);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,17)(13,19)(14,24)(15,22)(18,20)(21,23);(9,12)(10,16)(11,17)(13,19)(14,24)(15,22)(18,20)(21,23);(9,13,15,14)(10,17,18,23)(11,20,21,16)(12,19,22,24)Polycyclic presentation:
2^9|b^2=fg,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308423, #72308640, #72312072, … |
| maximal normal subgroup of | #72308423, #72308640, #72312072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353925);