Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12708 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #536 | C23 |
| prime radical | #61930 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3510 ◁ G | |
| LECS | G ▷ #536 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
539 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,15,16,47)(4,21,7,13)(5,24)(6,12)(8,14)(9,17,34,50)(10,23,38,30)(11,26,41,22)(18,46,51,64)(19,48,20,49)(25,44,52,35)(27,60)(28,57)(29,37)(31,39)(32,61,42,54)(33,53,36,63)(40,59,43,58)(45,56,62,55);(1,3)(2,9)(4,19)(5,25)(6,18)(7,16)(8,20)(10,36)(11,15)(12,42)(13,44)(14,35)(17,28)(21,34)(22,50)(23,55)(24,47)(26,52)(27,54)(29,53)(30,51)(31,56)(32,38)(33,39)(37,64)(40,62)(41,49)(43,46)(45,60)(48,57)(58,63)(59,61);(1,4,8,7)(2,10)(3,16,20,19)(5,26,28,22)(6,23,31,30)(9,32)(11,37)(12,21)(13,39)(14,38)(15,45)(17,52,25,50)(18,51,56,55)(24,40)(27,59,29,58)(33,34)(35,36)(41,60)(42,44)(43,57)(46,47)(48,62)(49,64)(53,61,54,63)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59344);