Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(7,12)(10,11)(13,14)(15,19)(16,18)(17,20);(1,3,4,2)(5,6,8,11)(7,10,12,9);(14,18)(15,16)(19,20);(6,10)(7,12)(9,11)(13,15,17,16)(14,19,20,18)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,d^c=di,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279250, #72279339, #72279546 |
| maximal normal subgroup of | #72279250, #72279339, #72279546 |
| maximal quotient of | #72309477, #72309601, #72309628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134540);