Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94823 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7529 |
| 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 | |
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:
(1,2,4,11,8,14,25,42)(3,15,16,39,20,46,51,63)(5,12,27,40,30,37,47,62)(6,13,24,44,7,31,23,41)(9,21,32,53,35,55,60,17)(10,22,36,29,38,56,57,28)(18,45,50,64,19,33,49,58)(26,34,48,61,52,43,54,59);(1,3,8,20)(2,9,14,35)(4,21,25,55)(5,26,30,52)(6,19,7,18)(10,33,38,45)(11,39,42,63)(12,15,37,46)(13,43,31,34)(16,47,51,27)(17,29,53,28)(22,49,56,50)(23,48,24,54)(32,57,60,36)(40,61,62,59)(41,64,44,58)Polycyclic presentation:
2^9|a^2=d,b^2=i,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=cfgh,d^b=def,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2022);