Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,15,21,24)(12,17)(13,19)(14,18)(16,20,23,22);(1,3,5,8,4,2,6,7)(9,11)(10,15,18,24)(12,20)(13,22)(14,21)(16,17,23,19);(2,3)(7,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,17)(18,23)(19,24);(9,12,14,13)(10,17,18,19)(11,20,21,22)(15,23,24,16)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103374);