Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 56 points:
(2,6,7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26,29,34,30,35,44,52,31,36,45,53,43,51,28,33)(27,37,39,49,40,50,38,54,41,55,42,32,46,47,56,48);(1,2)(3,6)(4,8)(5,7)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,32)(28,42)(29,38)(30,46)(31,41)(33,50)(34,37)(35,54)(36,49)(39,44)(40,43)(45,56)(47,52)(48,51)(53,55);(1,3,5,4)(2,6,7,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fghi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31669);