Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C2×C8 |
| Frattini, cosocle | C8×C16 | C22 |
| socle | C22 | #1331 |
| prime radical | C2×D16 | C8 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,7,5,8,10,14,4,6,9,12,11,13,15,16)(17,18,20,26,22,28,23,39)(19,29,32,46,34,24,35,40)(21,36,31,43,38,44,37,27)(25,42,41,30,45,48,47,33);(2,6)(3,9)(8,13)(10,15)(18,24)(19,30)(20,23)(21,37)(25,43)(26,46)(27,45)(28,29)(31,38)(32,42)(33,34)(35,48)(36,41)(39,40)(44,47)Polycyclic presentation:
2^9|a^2=d,c^2=eghi,d^2=f,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cehi,d^b=dhi,e^b=eg,g^b=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 960);