Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130498 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#547 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10)(11,21,25,32)(12,26,30,35)(13,19,14,18)(15,20)(16,23)(17,29)(22,34)(24,33)(27,31)(28,36)(37,38,40,39);(2,6)(3,4)(7,8)(9,11,15,25)(10,16,20,33)(12,27,30,22)(13,24,14,23)(17,34,36,31)(18,32,19,21)(26,37,35,40)(28,38,29,39);(9,12,15,30)(10,17,20,36)(11,22,25,27)(13,29,14,28)(16,31,33,34)(18,26,19,35)(21,37,32,40)(23,39,24,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72282351, #72282369, #72282538, … |
| maximal normal subgroup of | #72282351, #72282369, #72282538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37971);