Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(6,10)(7,11)(8,12)(13,14,19,25)(15,26,31,43)(16,32,36,41)(17,24,18,23)(20,27,39,28)(21,35,22,34)(29,40,30,42)(33,38,44,37);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,15,19,31)(14,20,25,39)(16,33,36,44)(17,29,18,30)(21,40,22,42)(23,37,24,38)(26,41,43,32)(27,35,28,34);(2,4)(5,7,9,6)(8,10,12,11)(13,16,18,35,19,36,17,34)(14,21,24,41,25,22,23,32)(15,27,29,44,31,28,30,33)(20,26,37,42,39,43,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=ehi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50487);