Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×AGL(1,ℤ/8) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10,14,18,15,17,13,11)(12,20,21,16,22,19,23,24);(1,3,5,8,4,2,6,7)(9,11,14,17,15,18,13,10)(12,20,23,16,22,19,21,24);(2,3)(7,8)(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,24)(18,20);(9,13,15,14)(10,11,17,18)(12,21,22,23)(16,19,24,20)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,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308938, C2×#196180 |
| maximal normal subgroup of | #72308938, C2×#196180 |
| maximal quotient of | C2×#196180, #72309121, #72309143 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103376);