Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #1368 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | #1311 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,33,26,35,38,36)(19,27,28,41)(20,30)(21,32)(25,39,37,34,40,47,46,48)(29,44,43,42)(31,45);(17,18,19,23,20,25,27,34,21,24,29,36,30,37,42,39,22,26,28,35,31,40,41,48,32,38,43,33,45,46,44,47)Polycyclic presentation:
2^9|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=dgi,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224639, #72224648, #72224654, … |
| maximal normal subgroup of | #72224639, #72224648, #72224654, … |
| maximal quotient of | #72224648, #72230695, C2×#93877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1073);