Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(11,17)(12,20)(14,22)(16,23)(18,24)(19,21);(1,2,4,3)(5,6)(7,8)(9,10,13,15)(11,18,19,14)(12,16,20,23)(17,24,21,22);(1,3,4,2)(6,8)(9,11)(10,14)(12,21)(13,19)(15,18)(16,24)(17,20)(22,23);(6,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,19)(20,23)(21,24);(10,15)(14,18)(16,23)(22,24)Polycyclic presentation:
2^9|b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315616, #72315640, #72315856, … |
| maximal normal subgroup of | #72315616, #72315640, #72315856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346022);