Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142544 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 655 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,17)(13,16)(14,19)(15,21)(18,26)(22,33)(23,35)(24,30)(25,32)(27,40)(28,36)(29,37)(31,39)(34,38);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,29)(17,20)(18,27)(19,28)(24,37)(25,36)(26,38)(30,33)(31,34)(32,35)(39,40);(3,7)(5,8)(10,16)(13,26)(15,27)(17,30)(18,19)(20,33)(21,36)(22,38)(24,39)(28,29)(31,32)(34,35)Polycyclic presentation:
2^9|d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,e^c=ei,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283973, #72284061, #72284113, … |
| maximal normal subgroup of | #72283973, #72284061, #72284113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49740);