Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #123111 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×C4⋊C8 | C23 |
| prime radical | #31737 | C2 |
| LCS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
339 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,7)(5,9)(8,13)(10,15)(11,16)(12,14)(17,18,23,29)(19,30,33,45)(20,24,37,41)(21,28,22,27)(25,38,43,32)(26,36,40,34)(31,44,35,42)(39,47,46,48);(1,3)(2,6)(4,12)(5,10)(7,16)(8,14)(9,15)(11,13)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,44)(31,47)(32,38)(35,48)(36,37)(42,45)(43,46);(2,7)(3,10)(4,13)(6,14)(8,9)(15,16)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,39)(26,44)(27,45)(28,30)(29,43)(31,37)(34,48)(36,47)(40,42)(41,46)Polycyclic presentation:
2^9|a^2=h,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30317);