Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138698 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,20,28,23,30,21,24,22,29,37,26,39,47,40,27)(19,31,33,25,35,43,34,45,32,46,38,44,41,42,36,48);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(18,24,29,27)(19,32)(20,21)(23,39)(25,43,44,42)(26,30,28,47)(31,48,46,45)(33,36)(34,38)(37,40);(3,11)(7,15)(10,16)(12,13)(17,19,22,32)(18,25,29,44)(20,36,37,34)(21,38,40,33)(23,41,39,35)(24,42,27,43)(26,46,28,31)(30,48,47,45)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285410, #72285438, #72285533, … |
| maximal normal subgroup of | #72285410, #72285438, #72285533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45963);