Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62788 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,14,18,15,11,13,17)(12,16,21,24,22,19,20,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23);(9,12,15,22)(10,16,11,19)(13,21,14,20)(17,24,18,23);(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72268183, #72268189, #72268190, … |
| maximal normal subgroup of | #72268183, #72268189, #72268190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93087);