Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
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 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
514 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 514 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(5,6)(7,9)(8,11)(10,12)(13,14,19,24)(15,25,16,28)(17,31,36,44)(18,37,34,23)(20,29,21,26)(22,40,43,35)(27,42,30,41)(32,39,33,38);(1,2)(3,4)(6,9)(7,12)(10,11)(13,15,19,16)(14,20,24,21)(17,32,36,33)(18,30,34,27)(22,41,43,42)(23,39,37,38)(25,44,28,31)(26,40,29,35);(2,4)(13,16,19,15)(14,21,24,20)(17,33,36,32)(18,27,34,30)(22,42,43,41)(23,38,37,39)(25,31,28,44)(26,35,29,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,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, 50426);