Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,33,39)(16,25,19,23)(17,22,18,37)(21,32,40,29)(24,34,41,35)(28,42,36,43)(30,38,31,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,21)(16,32)(17,36)(18,28)(19,29)(20,33)(22,38)(23,39)(24,42)(25,27)(26,40)(30,35)(31,34)(37,44)(41,43);(1,3,4,2)(9,12)(14,22)(15,28)(16,19)(17,35)(18,34)(23,41)(24,25)(26,37)(27,39)(29,30)(31,32)(33,36)(38,42)(43,44)Polycyclic presentation:
2^9|a^2=i,c^2=dg,d^2=i,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=eg,f^c=fg,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52125);