Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #412 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2450 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)(9,10,15,18,14,20,13,19)(11,16,21,23,17,12,22,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,17)(15,22)(18,24)(19,23);(9,12)(10,17)(11,20)(13,23)(14,16)(15,24)(18,22)(19,21);(9,13,14,15)(10,18,20,19)(11,21,17,22)(12,23,16,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322064, #72322177, #72322190 |
| maximal normal subgroup of | #72322064, #72322177, #72322190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93763);