Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#386 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8,5,3,4,7)(10,14,17,20)(11,18,19,21);(1,3,5,2)(4,7,6,8)(9,10,15,14,13,17,12,20)(11,16,18,22,19,23,21,24);(4,6)(7,8)(9,11)(10,16)(12,21)(13,19)(14,22)(15,18)(17,23)(20,24);(10,17)(14,20)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102987);