Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| 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×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
828 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,7)(8,10,12,11)(14,17)(15,19)(18,20);(1,3,4,2)(6,10)(7,11)(8,12);(6,7)(10,11)(13,14,16,18)(15,20,19,17);(5,7,9,6)(8,11,12,10)(14,18)(17,20)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dhi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279776, #72280072, #72309175 |
| maximal normal subgroup of | #72279776, #72280072, #72309175 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134248);