Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152563 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13041 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #33569 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 64 points:
(1,2,6,11,8,14,5,28)(3,15,19,36,21,46,17,9)(4,22,25,10,27,37,23,39)(7,13,30,43,33,41,31,12)(16,34,48,60,50,63,47,58)(18,53,52,42,54,35,20,56)(24,44,26,40,29,38,32,59)(45,49,62,51,61,57,64,55);(1,3)(2,9,14,36)(4,16)(5,17)(6,19)(7,20)(8,21)(10,34,39,63)(11,15,28,46)(12,42,43,56)(13,35,41,53)(18,30)(22,58,37,60)(23,47)(24,55)(25,48)(26,49)(27,50)(29,51)(31,54)(32,57)(33,52)(38,62,44,64)(40,45,59,61);(1,4)(2,10)(3,16)(5,29)(6,24)(7,32)(8,27)(9,34)(11,40)(12,37)(13,44)(14,39)(15,45)(17,51)(18,47)(19,55)(20,57)(21,50)(22,43)(23,30)(25,31)(26,33)(28,59)(35,64)(36,63)(38,41)(42,60)(46,61)(48,54)(49,52)(53,62)(56,58)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59761);