Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,14,19,15,20,11,21)(12,18,22,17,13,16,23,24);(2,4)(5,6,7,8)(10,16)(12,13)(17,19)(18,20)(21,24)(22,23);(9,11,15,14)(10,17,20,24)(12,23,13,22)(16,19,18,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303499 |
| maximal normal subgroup of | #72303499 |
| maximal quotient of | #72303595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53917);