Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124851 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,6)(7,8)(9,10,13,20,16,22,32,34,14,23,30,17,15,21,12,19)(11,24,26,35,29,38,25,33,27,37,40,39,28,18,31,36);(2,5)(3,8)(6,7)(10,17)(12,30)(13,32)(15,16)(18,33)(19,21)(20,23)(22,34)(24,35)(25,26)(28,29)(31,40)(36,38)(37,39);(2,6)(5,7)(9,11)(10,18,23,38)(12,31)(13,26)(14,27)(15,28)(16,29)(17,33,20,36)(19,39,34,35)(21,37,22,24)(25,32)(30,40)Polycyclic presentation:
2^9|a^2=e,c^2=fghi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240791, #72240794, #72240797, … |
| maximal normal subgroup of | #72240791, #72240794, #72240797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32111);