Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152130 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12698 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3509 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,7,13,26,41)(3,14,17,44,16,42,47,39)(4,19,22,49,21,9,34,36)(6,12,28,40,29,10,25,38)(8,30,33,18,32,24,57,27)(15,43,45,63,20,50,51,61)(23,35,56,60,53,37,55,62)(31,54,59,46,48,58,64,52);(1,3,7,16)(2,8)(4,20,21,15)(5,24,26,30)(6,27,29,18)(9,31)(10,14)(11,39)(12,42)(13,32)(17,25,47,28)(19,48)(22,54,34,58)(23,46,53,52)(33,38)(35,43)(36,61)(37,50)(40,57)(41,44)(45,56,51,55)(49,63)(59,60)(62,64);(1,4,7,21)(2,9)(3,15,16,20)(5,22,26,34)(6,23,29,53)(8,31)(10,35)(11,36)(12,37)(13,19)(14,43)(17,45,47,51)(18,46,27,52)(24,58,30,54)(25,55,28,56)(32,48)(33,59)(38,60)(39,61)(40,62)(41,49)(42,50)(44,63)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59327);