Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94819 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7515 |
| commutator | #493 | C2×C4 |
| Frattini, cosocle | #1712 | C22 |
| prime radical | #12310 | C2 |
| LCS | G ▷ #493 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1712 ◁ G | |
| LECS | G ▷ #1712 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 128 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)(3,13,14,8)(4,10,18,9)(5,11,6,22)(7,12)(15,29,32,23)(16,30,31,25)(17,24,21,28)(19,27,20,26);(1,3)(2,8)(4,17)(5,21)(6,15)(7,16)(9,13)(10,28)(11,24)(12,25)(14,18)(19,31)(20,32)(22,23)(26,29)(27,30)Polycyclic presentation:
2^9|a^2=d,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,d^b=deghi,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, 2015);