Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143943 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| 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 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 470 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:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,18,22)(15,23,26,42)(16,28,31,41)(17,32,30,21)(19,34,36,25)(20,38,39,29)(24,40,43,37)(27,44,33,35);(2,4)(13,15,18,26)(14,19,22,36)(16,24,31,43)(17,33,30,27)(20,35,39,44)(21,40,32,37)(23,41,42,28)(25,38,34,29);(6,9)(7,12)(10,11)(15,24)(16,29)(17,30)(19,35)(20,28)(21,32)(23,37)(25,27)(26,43)(31,38)(33,34)(36,44)(39,41)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51142);