Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153673 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13705 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| prime radical | #2262 | C22 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
97 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(2,10)(3,15,19,49)(4,8)(5,22)(6,28)(7,14)(11,36,18,52)(12,39)(13,30)(16,50,33,46)(17,40,51,58)(20,53)(21,44)(23,29,47,38)(24,31)(25,59)(26,55)(27,45)(32,48,34,37)(35,41)(42,57)(43,60,62,64)(54,63,61,56);(1,2,8,7,9,14,4,10)(3,11,33,32,19,18,16,34)(5,12,26,25,27,42,20,35)(6,13,30,28,31,44,21,24)(15,37,50,52,49,48,46,36)(17,38,62,56,51,29,43,63)(22,41,53,57,45,59,55,39)(23,40,61,60,47,58,54,64);(1,3)(2,11)(4,16)(5,23)(6,29)(7,32)(8,33)(9,19)(10,34)(12,40)(13,43)(14,18)(15,45)(17,28)(20,54)(21,56)(22,49)(24,51)(25,60)(26,61)(27,47)(30,63)(31,38)(35,64)(36,57)(37,59)(39,52)(41,48)(42,58)(44,62)(46,53)(50,55)Polycyclic presentation:
2^9|a^2=dh,b^2=di,d^2=i,e^2=fgi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ef,f^a=fg,g^a=gh,h^a=hi,e^c=efi,f^c=fg,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72225423, #72225424, #72225425, … |
| maximal normal subgroup of | #72225423, #72225424, #72225425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60870);