Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13605 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2327 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,12,18,14,19,24,33)(11,16,21,30,20,31,34,40)(13,17,23,32,26,15,25,28)(22,29,36,39,35,27,37,38);(10,15)(12,23)(13,26)(16,29)(17,19)(21,36)(22,35)(24,25)(27,31)(28,32)(30,40)(34,37);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,29)(18,30)(19,31)(23,36)(24,34)(25,37)(26,35)(28,38)(32,39)(33,40)Polycyclic presentation:
2^9|a^2=e,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dh,e^b=efhi,f^b=fi,d^c=di,e^d=ehi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72292986, #72292987, #72292988, … |
| maximal normal subgroup of | #72292986, #72292987, #72292988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56908);