Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94674 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7412 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1960 | C22 |
| prime radical | C23 | #354 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1960 ◁ G | |
| LECS | G ▷ #1960 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,13,27,44)(3,14,16,46,18,48,53,64)(4,19,21,42,24,39,56,62)(7,30,29,43,8,12,28,45)(9,32,33,15,36,49,41,50)(10,26,38,22,40,55,63,23)(17,47,25,57,31,37,52,58)(20,35,51,59,54,34,61,60);(1,3,8,17,6,18,7,31)(2,9,12,34,13,36,30,35)(4,20,23,50,24,54,22,15)(5,25,29,16,27,52,28,53)(10,37,39,14,40,47,19,48)(11,41,43,60,44,33,45,59)(21,49,55,61,56,32,26,51)(38,58,42,64,63,57,62,46)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fhi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gh,h^a=hi,c^b=cf,d^b=defi,e^b=eh,f^b=fi,d^c=dgh,e^c=ei,e^d=ehi,f^d=fi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1871);