Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,16,18,11)(9,15)(10,12)(14,17,20,19);(5,7,13,6,10,15,9,12)(8,14,17,11,18,20,19,16);(5,8)(6,11)(7,14)(9,19)(10,18)(12,16)(13,17)(15,20);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214896, #72224746, #72224752, … |
| maximal normal subgroup of | #72214896, #72224746, #72224752, … |
| maximal quotient of | #72213628, #72213768, #72214417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299891);