Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2000 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,14)(8,9)(12,19);(2,4)(5,6,9,13,10,11,8,18)(7,15,17,19,14,20,16,12);(1,2,3,4)(5,7,8,16,10,14,9,17)(6,12,13,15,11,19,18,20);(12,19)(15,20)Polycyclic presentation:
2^9|b^2=h,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,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 | #72238972, #72238975 |
| maximal normal subgroup of | #72238972, #72238975 |
| maximal quotient of | #72238983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299230);