Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94830 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7530 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | #1713 | C22 |
| prime radical | C22 | #1408 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1713 ◁ G | |
| LECS | G ▷ #1713 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,8,14,25,43)(3,15,17,47,16,44,48,40)(5,12,27,41,30,38,57,64)(6,13,24,46,7,31,23,42)(9,32,33,18,36,21,55,49)(10,22,37,29,39,58,59,28)(19,34,52,62,20,54,51,60)(26,45,50,61,53,35,56,63);(1,3)(2,9,14,36)(4,21)(5,26)(6,20)(7,19)(8,16)(10,34,39,54)(11,40,43,47)(12,44,38,15)(13,45,31,35)(17,27)(18,28)(22,51)(23,50)(24,56)(25,32)(29,49)(30,53)(33,59,55,37)(41,63,64,61)(42,60,46,62)(48,57)(52,58)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fg,d^2=i,e^2=gh,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cfghi,d^b=def,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2027);