Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499266 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,24)(13,18)(14,16)(17,26)(19,35)(20,34)(21,30)(22,38)(23,29)(25,39)(27,32)(28,37)(31,36)(33,40);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,30)(17,28)(18,34)(22,25)(23,27)(24,35)(26,37)(29,32)(31,33)(36,40)(38,39);(1,3,5,4)(2,7,8,6)(11,19)(12,25)(13,27)(15,28)(17,33)(18,32)(20,36)(21,37)(22,23)(24,39)(26,40)(29,31)(30,35)(34,38);(1,4,5,3)(2,6,8,7)(10,16)(11,20,21,23)(12,26)(15,29,30,34)(18,32)(19,36,37,22)(25,40)(28,31,35,38)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gh,h^b=hi,g^c=gh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239800, #72239811, #72239836, … |
| maximal normal subgroup of | #72239800, #72239811, #72239836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406906);