Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6,8,11)(7,9,12,10)(13,14)(15,27)(16,21)(17,23)(18,25)(19,24)(20,26)(22,29)(28,41)(30,36)(31,39)(32,43)(33,37)(34,38)(35,40)(42,44);(1,2)(3,4)(6,9)(7,12)(10,11)(14,21)(16,17)(18,19)(22,36)(23,24)(25,37)(26,38)(27,41)(28,44)(29,35)(30,31)(33,34)(39,42)(40,43);(2,4)(13,15,20,32)(14,22,26,42)(16,29,34,44)(17,35,33,28)(18,31,19,30)(21,36,38,39)(23,27,37,43)(24,41,25,40)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50599);