Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94743 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7451 |
| commutator | C2×C32 | C2×C4 |
| Frattini, cosocle | C4×C32 | C22 |
| prime radical | C22 | #1419 |
| LCS | G ▷ C2×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,12,26,48,57,10,37,38,24,56,6,33,27,62,5,30,22,55,19,53,59,49,35,51,23,46,31,50,25,54,9,16,29,45,61,47,58,42,7,15,3,17,20,14,52,41,8,13,28,40,34,43,18,39,32,63,21,11,36,44,60,64);(2,10,16,42)(3,18)(4,25)(5,8)(6,34)(7,35)(11,14,46,33)(12,47,45,48)(13,49,30,39)(15,54,38,64)(17,44,56,50)(19,20)(21,58)(22,27)(23,57)(24,59)(26,36)(28,52)(29,60)(31,61)(32,37)(40,43,55,53)(41,63,62,51)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fgi,d^2=efg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ei,f^a=fi,c^b=cf,d^b=def,e^b=ei,f^b=fg,g^b=gh,h^b=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72225423, #72225425, #72225427 |
| maximal normal subgroup of | #72225423, #72225425, #72225427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1939);