Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1347 |
| commutator | C22×C16 | C2×C4 |
| Frattini, cosocle | C2×C4⋊C16 | C22 |
| prime radical | C23 | #357 |
| LCS | G ▷ C22×C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C2×C4⋊C16 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
118 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,8,5,7,12,15)(3,9,10,16,11,6,13,14)(17,18)(19,23)(20,32)(21,35)(22,27)(24,28)(25,29)(26,33)(30,42)(31,38)(34,48)(36,44)(37,43)(39,47)(40,45)(41,46);(1,3,5,11)(2,6,7,9)(4,10,12,13)(8,16,15,14)(18,23)(19,28)(20,29)(21,33)(24,25)(26,42)(27,38)(30,47)(31,45)(32,36)(34,46)(35,37)(39,48)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=d,b^2=ghi,c^2=fh,d^2=ghi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,h^b=hi,d^c=dghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1055);