Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94678 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7416 |
| commutator | #423 | C2×C4 |
| Frattini, cosocle | #1838 | C22 |
| prime radical | C23 | #355 |
| LCS | G ▷ #423 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1838 ◁ G | |
| LECS | G ▷ #1838 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 122 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:
(1,2,5,10,6,12,30,42,7,13,29,43,22,11,44,45)(3,14,16,46,18,47,27,58,19,48,52,64,17,37,53,63)(4,20,23,55,25,9,36,38,26,39,57,41,24,40,28,59)(8,31,32,54,34,51,60,15,35,49,61,21,33,56,62,50);(1,3,7,19)(2,8,13,35)(4,21,26,54)(5,27,29,53)(6,18,22,17)(9,37,40,47)(10,32,43,61)(11,34,12,33)(14,39,48,20)(15,25,50,24)(16,30,52,44)(23,49,57,31)(28,51,36,56)(38,58,59,63)(41,46,55,64)(42,62,45,60)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ei32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1875);