Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94737 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7444 |
| commutator | C2×C32 | C2×C4 |
| Frattini, cosocle | C4×C32 | C22 |
| prime radical | #13725 | C2 |
| 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 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,12,8,37,7,35,9,17,29,15,36,16,26,52)(3,18,21,55,24,44,23,42,25,43,60,50,61,10,38,39)(5,30,27,14,32,46,31,13,33,51,20,54,6,34,28,53)(11,45,40,58,64,62,56,63,49,59,47,19,48,57,41,22);(2,10)(3,19)(4,26)(5,31)(6,20)(7,29)(8,36)(11,14)(12,50)(13,48)(15,55)(16,18)(17,44)(21,59)(22,61)(23,62)(24,63)(25,58)(27,32)(28,33)(30,40)(34,56)(35,42)(37,43)(38,57)(39,52)(41,46)(45,60)(47,51)(49,54)(53,64)Polycyclic presentation:
2^9|a^2=d,c^2=fgi,d^2=efgi,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fgi,g^b=gh,h^b=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72225422, #72225425, #72225427 |
| maximal normal subgroup of | #72225422, #72225425, #72225427 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1934);