Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143815 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19258 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 48 points:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,15)(11,16)(13,14)(17,18)(19,27)(20,31)(21,36)(22,25)(23,29)(24,34)(26,33)(28,43)(30,45)(32,42)(35,46)(37,40)(38,44)(39,47)(41,48);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(23,38)(27,43)(39,42)(40,45);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(17,19)(18,23)(20,32)(21,37)(22,28)(24,40)(25,38)(26,42)(27,31)(29,34)(30,33)(35,47)(36,39)(41,45)(43,46)(44,48)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,d^c=dgi,e^c=eh,h^c=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51035);