Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 538 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) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,22)(18,24)(21,23)(25,26)(27,37)(28,42)(29,34)(30,35)(31,36)(32,41)(33,43)(38,46)(39,48)(40,47)(44,50)(45,49)(51,53)(52,56)(54,55);(2,5)(6,8)(9,11,13,17)(10,14,15,22)(12,18,20,23)(16,21,19,24)(25,27,29,40,31,39,30,41)(26,32,35,48,36,47,34,37)(28,38,43,53,45,52,44,54)(33,46,42,55,50,56,49,51);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,19)(18,23)(25,28,30,44,31,45,29,43)(26,33,34,49,36,50,35,42)(27,38,41,54,39,52,40,53)(32,46,37,51,47,56,48,55);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,21,22,24)(27,39)(32,47)(37,48)(38,52)(40,41)(46,56)(51,55)(53,54);(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,43,45,44)(32,37,47,48)(33,49,50,42)(38,53,52,54)(46,51,56,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343533);