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, 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 | |
422 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 422 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(6,9)(7,11)(10,12)(13,14,18,23)(15,24,27,42)(16,19,31,35)(17,32,30,22)(20,36,38,25)(21,39,34,28)(26,43,33,40)(29,41,44,37);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(14,19)(16,28)(17,30)(20,24)(21,22)(23,35)(25,29)(26,33)(31,39)(32,34)(36,44)(37,43)(38,42)(40,41);(2,4)(13,15,18,27)(14,20,23,38)(16,29,31,44)(17,33,30,26)(19,24,35,42)(21,40,34,43)(22,41,32,37)(25,39,36,28)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,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, 50504);