Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13605 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2328 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 357 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)(11,19,21,34)(12,23,24,39)(13,26,28,17)(14,18)(15,20,30,37)(16,25)(22,32,27,35)(29,36,31,38)(33,40);(9,11)(10,15)(12,20)(13,27)(14,22)(16,32)(17,30)(18,29)(19,33)(21,28)(23,35)(24,36)(25,37)(26,31)(34,39)(38,40);(1,3,5,4)(2,6,8,7)(12,24)(15,29)(16,33)(20,36)(23,39)(25,40)(30,31)(37,38)Polycyclic presentation:
2^9|a^2=e,c^2=g,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 | #72292972, #72292975, #72292976, … |
| maximal normal subgroup of | #72292972, #72292975, #72292976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56907);