Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5284 conjugacy classes of subgroups; only the 610 normal subgroups are shown.
This group has 5284 conjugacy classes of subgroups — too many to draw. Only its 610 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(14,17)(15,18)(19,20);(10,12)(13,14,18,20,16,19,15,17);(2,4);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei152 conjugacy classes — view the full character table.
| Fitting subgroup of | #72341794 |
| maximal subgroup of | #72237125, #72237127, #72237129, … |
| maximal normal subgroup of | #72237125, #72237127, #72237129, … |
| maximal quotient of | #72249972, #72250254, #72256958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299863);