Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,3,7,8)(9,10,15,17,14,12,13,21)(11,16,19,24,20,18,22,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,16)(12,18)(13,19)(14,20)(15,22)(17,23)(21,24);(4,7)(6,8)(9,12)(10,14)(11,18)(13,21)(15,17)(16,20)(19,24)(22,23);(9,13,14,15)(10,17,12,21)(11,19,20,22)(16,23,18,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95098);