Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,11,12,7);(5,6,9,11,10,12,8,7)(13,14)(15,17)(16,19)(18,20);(5,7,8,6,10,11,9,12)(14,17)(15,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240042 |
| maximal normal subgroup of | #72240042 |
| maximal quotient of | #72240043 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27823);