Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2258 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2258 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:
(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20);(2,4)(5,7)(6,10,12,13)(8,14)(9,15)(11,19,16,18)(17,20);(1,2,3,4)(5,8)(6,11,12,16)(7,14)(9,17)(10,18,13,19)(15,20);(14,20)(18,19)Polycyclic presentation:
2^9|b^2=h,c^2=g,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225667, #72225682, #72225685, … |
| maximal normal subgroup of | #72225667, #72225682, #72225685, … |
| maximal quotient of | #72238983, #72240089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299226);