Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1189 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1189 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11,15,16)(8,12)(9,13,18,20)(10,14)(17,19);(1,2)(3,4)(5,7)(6,11,14,16)(8,13,17,20)(9,12)(10,15)(18,19);(2,4)(5,8)(6,12)(7,13,15,20)(9,16,18,11)(10,17)(14,19);(5,9)(6,13,14,20)(7,12)(8,16,17,11)(10,18)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267756 |
| maximal normal subgroup of | #72267756 |
| maximal quotient of | #72267759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385444);