Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#459 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,10)(7,11,12,8)(14,17)(15,18)(19,20);(1,3,4,2)(6,7)(8,11)(10,12)(13,14,16,17)(15,18,19,20);(6,10)(7,12)(13,15,16,19)(14,18,17,20);(5,7)(6,11)(8,10)(9,12)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dh,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279848, #72279954, #72280072, … |
| maximal normal subgroup of | #72279848, #72279954, #72280072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134770);