Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143219 |
| 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 | |
387 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 387 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:
(2,5)(3,7)(6,8)(9,10,15,21)(11,22,26,35)(12,16,29,32)(13,20,14,19)(17,23,36,25)(18,28,31,27)(24,34,39,33)(30,38,37,40);(1,2,4,6)(3,8,7,5)(10,16)(11,23)(12,27)(13,14)(17,33)(18,19)(20,31)(21,32)(22,37)(24,39)(25,26)(28,29)(30,35)(34,36);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,24)(13,23)(14,25)(15,26)(16,30)(18,34)(19,35)(20,22)(21,36)(27,38)(28,40)(29,39)(31,33)(32,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50477);