Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3169 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
276 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,8)(4,7)(5,14)(6,10)(9,12)(11,15)(13,16)(17,18,20,25,21,26,30,38,22,24,31,36,32,37,42,46)(19,23,28,33,29,34,41,43,27,35,39,44,40,45,47,48);(2,7)(3,5)(4,12)(8,9)(10,16)(11,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,35)(25,33)(26,34)(30,41)(31,39)(32,40)(36,44)(37,45)(38,43)(42,47)(46,48);(1,3,6,5)(2,8,10,14)(4,11,13,12)(7,15,16,9)(18,24)(19,27)(25,36)(26,37)(28,39)(29,40)(38,46)(41,47)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=fh,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=chi,d^b=df,f^b=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29656);