Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C23 | #357 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,3,5,4,6,7,8)(9,10)(11,22,25,52)(12,18)(13,30)(14,35)(15,39)(16,21)(17,24,41,56)(19,32)(20,37)(23,50,27,61)(26,60,59,72)(28,62)(29,49)(31,48,57,51)(33,70)(34,66)(36,53,55,45)(38,46)(40,54,43,58)(42,67,71,63)(44,64)(47,69)(65,68);(9,11,16,25)(10,17,21,41)(12,27,15,23)(13,31,34,57)(14,36,38,55)(18,43,39,40)(19,45,47,53)(20,48,49,51)(22,30,52,66)(24,32,56,69)(26,29,59,37)(28,63,33,67)(35,42,46,71)(44,60,65,72)(50,70,61,62)(54,68,58,64);(9,12,16,15)(10,18,21,39)(11,23,25,27)(13,28,34,33)(14,29,38,37)(17,40,41,43)(19,44,47,65)(20,35,49,46)(22,50,52,61)(24,54,56,58)(26,55,59,36)(30,62,66,70)(31,67,57,63)(32,64,69,68)(42,48,71,51)(45,72,53,60)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dfi,e^b=ei,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232001, #72232003, #72234342, … |
| maximal normal subgroup of | #72232001, #72232003, #72234342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57261);