Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143942 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,6,3)(4,7,5,8)(9,10,15,21)(11,23,12,26)(13,22,16,19)(14,30,32,38)(17,27,35,24)(18,25,37,28)(20,29,34,33)(31,39,40,36);(2,7)(3,8)(4,5)(9,11)(10,17)(12,15)(13,27)(14,28)(16,24)(18,34)(19,26)(20,37)(21,35)(22,23)(25,32)(29,40)(30,39)(31,33)(36,38);(1,3,6,2)(4,8,5,7)(9,12,15,11)(10,18,21,37)(13,27,16,24)(14,31,32,40)(17,34,35,20)(19,36,22,39)(23,30,26,38)(25,33,28,29)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fgi,d^c=dgi,e^c=eh,f^c=fgi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51216);