Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28491 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 542 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) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,10)(8,11)(9,12)(13,14,18,22)(15,23,25,41)(16,27,29,43)(17,32,30,21)(19,34,35,24)(20,37,39,31)(26,42,33,38)(28,40,44,36);(1,2)(3,4)(6,10)(7,8)(11,12)(13,15)(14,19)(16,28)(17,33)(18,25)(20,38)(21,40)(22,35)(23,27)(24,31)(26,30)(29,44)(32,36)(34,37)(39,42)(41,43);(2,4)(6,11)(7,8)(10,12)(16,29)(20,39)(23,41)(24,34)(27,43)(28,44)(31,37)(38,42)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,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, 50411);