Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500599 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 733 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(6,9)(7,12)(8,10)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(5,6,8,9,11,14,10,13)(7,15,17,20,18,19,16,12);(2,4)(5,7)(6,12)(8,16)(9,19)(10,17)(11,18)(13,15)(14,20);(5,6,10,13,11,14,8,9)(7,12,17,15,18,20,16,19)Polycyclic presentation:
2^9|b^2=g,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265494, #72265587, #72265615, … |
| maximal normal subgroup of | #72265494, #72265587, #72265615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413498);