Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11)(3,13,18,34)(4,9,19,31)(5,22,26,10)(7,12,14,27)(8,23,30,17)(15,36,43,60)(16,37,38,62)(20,48,47,32)(21,33,39,56)(24,53,54,61)(25,50,40,35)(28,49,59,42)(29,52,55,45)(41,63,44,64)(46,58,51,57);(1,3,14,38)(2,8,27,55)(4,15,39,41)(5,23,40,45)(6,18,7,16)(9,28,56,57)(10,34,50,37)(11,30,12,29)(13,35,62,22)(17,25,52,26)(19,43,21,44)(20,49,54,46)(24,51,47,42)(31,59,33,58)(32,60,53,64)(36,61,63,48);(1,4,7,21)(2,9,12,33)(3,15,16,44)(5,24,25,20)(6,19,14,39)(8,28,29,58)(10,32,35,61)(11,31,27,56)(13,36,37,64)(17,46,45,42)(18,43,38,41)(22,48,50,53)(23,51,52,49)(26,54,40,47)(30,59,55,57)(34,60,62,63);(3,16)(4,19)(8,29)(9,31)(13,37)(15,41)(17,45)(18,38)(20,47)(21,39)(23,52)(24,54)(28,57)(30,55)(32,48)(33,56)(34,62)(36,63)(42,51)(43,44)(46,49)(53,61)(58,59)(60,64)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394386);