Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | #2215 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×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 | |
202 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 202 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) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 80 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,27,43,59,40,56,23,29,41,57,22,28,44,60,24,30,45,61,38,54,73,79,47,63,20,26,39,55,74,80)(19,31,33,50,69,58,72,52,35,65,32,62,34,53,42,64,36,66,67,77,46,75,37,25,48,49,68,76,70,78,71,51);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,25)(20,37)(21,42)(22,46)(23,48)(24,36)(26,53)(27,58)(28,62)(29,64)(30,52)(31,56)(32,40)(33,44)(34,38)(35,47)(39,69)(41,72)(43,70)(45,71)(49,60)(50,54)(51,63)(55,76)(57,78)(59,77)(61,75)(65,80)(66,79)(67,74)(68,73);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58169);