Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94821 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7515 |
| commutator | #494 | C2×C4 |
| Frattini, cosocle | #1713 | C22 |
| prime radical | #3293 | C4 |
| LCS | G ▷ #494 ▷ 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 | |
114 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,8,3,13,15)(5,11,22,9,25,16,32,14)(6,12,20,30,7,17,19,28)(18,27,24,21,31,26,23,29);(1,3)(2,8)(4,9)(5,21)(6,12)(7,17)(10,22)(11,24)(13,14)(15,32)(16,23)(18,28)(19,27)(20,26)(25,29)(30,31)Polycyclic presentation:
2^9|a^2=d,c^2=i,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=ci,d^b=degh,e^b=eghi,f^b=fgh,h^b=hi,d^c=dfg,e^c=eghi,f^c=fhi,g^c=gi,e^d=eghi,f^d=fhi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2018);