Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | C2×#322 | C22 |
| prime radical | C23 | #329 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×#322 ▷ C2×C42 ▷ C22 ▷ C1 | |
149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,10,6,11,3,13,7,12,14,8,5,9,15,16)(17,18)(19,27,29,44,31,45,30,39)(20,32,35,24)(21,26)(22,25)(23,28,37,33,38,47,42,48)(34,40)(36,43,46,41);(1,3,7,15)(2,8,12,10)(4,5,14,6)(9,16,11,13)(17,19,22,31)(18,23,25,38)(20,33,35,48)(21,29,34,30)(24,39,32,44)(26,42,40,37)(27,43,45,41)(28,46,47,36)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fhi,d^2=g,e^2=i,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fhi,g^b=gh,d^c=dhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57);