Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #60695 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,4,7,5,3,8,6)(9,10)(11,21,24,36)(12,26)(13,18)(14,20)(15,22,32,39)(16,23,33,25)(17,28)(19,30)(27,37,40,31)(29,34)(35,38);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,27)(13,23)(14,24)(16,17)(18,31)(19,37)(20,32)(21,34)(22,38)(25,30)(26,36)(28,39)(29,40)(33,35);(2,3)(6,7)(9,11)(10,16)(12,22)(13,25)(14,24)(15,17)(18,36)(19,21)(20,33)(23,30)(26,31)(27,38)(28,40)(29,39)(32,35)(34,37)Polycyclic presentation:
2^9|a^2=f,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,f^b=fi,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319899, #72319904, #72319973, … |
| maximal normal subgroup of | #72319899, #72319904, #72319973, … |
| maximal quotient of | C2×#129565 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36761);