Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2842 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14636 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,21)(3,12,15,24,14,8,22,25)(4,16,20,31,18,9,19,27)(13,28,17,26,29,23,32,30);(1,3,5,22,7,14,6,15)(2,8,10,25,11,12,21,24)(4,17,19,13,18,32,20,29)(9,26,27,23,16,30,31,28);(1,4,5,19,7,18,6,20)(2,9,10,27,11,16,21,31)(3,13,15,17,14,29,22,32)(8,23,24,26,12,28,25,30);(13,29)(17,32)(23,28)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72251919, #72251920, #72251930, … |
| maximal normal subgroup of | #72251919, #72251920, #72251930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298713);