Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C2×C42 | C2×C8 |
| Frattini, cosocle | #1831 | C22 |
| socle | C22 | #1324 |
| prime radical | C23 | #324 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1831 ◁ G | |
| LECS | G ▷ #1831 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,9,5,10,15,23,6,8,14,20,16,21,27,30)(3,7,12,17,13,19,26,28,11,18,24,22,25,29,32,31)(33,34)(35,43,45,56)(36,47,48,63)(37,50,51,41)(38,42)(39,44,52,59)(40,49)(46,58,62,57)(53,60,54,61)(55,64);(1,3)(2,7,8,18)(4,12)(5,13)(6,11)(9,22,20,17)(10,19,21,29)(14,24)(15,26)(16,25)(23,31,30,28)(27,32)(33,35,37,45,38,46,51,62)(34,39,41,53,42,54,50,52)(36,44,49,60,48,61,64,59)(40,43,47,57,55,58,63,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=gh,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=deghi,d^c=dgi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 932);