Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93866 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7096 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | C8×C16 | C22 |
| prime radical | C22 | #1357 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,10,5,9,21,34,25,13,43,40,51,35,47,58,7,14,23,41,27,36,55,62,6,12,22,39,26,11,42,38)(3,15,16,44,17,32,48,60,28,45,57,64,56,33,54,61,19,46,50,63,52,31,53,59,18,30,49,37,24,8,20,29);(1,3,7,19)(2,8,14,33)(4,20,23,54)(5,24,27,56)(6,28,25,18)(9,30,36,45)(10,37,41,64)(11,15,35,46)(12,32,13,31)(16,47,50,42)(17,51,52,26)(21,49,55,57)(22,48,43,53)(29,58,61,38)(34,59,62,60)(39,44,40,63)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=fh,d^2=efh,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fh,g^b=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72225579, #72225581, #72225584, … |
| maximal normal subgroup of | #72225579, #72225581, #72225584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1063);