Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1281 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C4×M32 | C22 |
| socle | C2 | #7196 |
| prime radical | C23 | #347 |
| LCS | G ▷ C2×C8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,14,28,44,7,12,29,45,24,10,26,38,8,13,27,43,31,39,59,64,22,37,55,42,4,21,23,41)(3,15,17,48,18,9,32,33,19,36,25,57,16,34,51,62,20,49,54,60,30,35,56,61,50,46,58,63,52,47,53,40);(1,3)(2,9,12,34,13,35,37,47)(4,16)(5,25,27,58)(6,30)(7,19)(8,20)(10,15,39,36,21,49,14,46)(11,40,42,61,43,62,45,33)(17,29,54,55)(18,31)(22,50)(23,56,26,32)(24,52)(28,53,59,51)(38,63,44,60,41,57,64,48)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fi,d^b=dfh,g^b=gi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286048, #72286149, #72286160, … |
| maximal normal subgroup of | #72286048, #72286149, #72286160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1316);