Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,16)(12,15)(13,22)(14,18)(17,20)(19,23)(21,24);(2,4)(5,6)(7,8)(10,15)(11,13)(12,20)(16,17)(18,24)(19,21)(22,23);(6,8)(10,15)(11,13)(12,20)(16,17)(18,24)(19,21)(22,23);(9,11,14,13)(10,16,18,22)(12,19,21,20)(15,23,24,17)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237254, #72237283, #72238789, … |
| maximal normal subgroup of | #72237254, #72237283, #72238789, … |
| maximal quotient of | #72237263, #72280142, C2×#498853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406049);