Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144059 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1677 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 330 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12)(7,18)(8,23)(9,27)(10,26)(13,31)(14,24)(15,16)(17,28)(19,33)(20,25)(22,36)(30,32)(34,35);(1,2)(3,4)(5,6)(7,19)(8,24)(9,28)(10,14)(11,17)(12,23)(13,21)(15,27)(16,26)(18,30)(20,34)(22,35)(25,33)(29,32)(31,36);(2,4)(5,7,8,25,26,36,9,29,11,22,27,20,10,18,23,21)(6,13,28,31,14,34,24,19,17,32,12,30,16,33,15,35)Polycyclic presentation:
2^9|c^2=dh,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,d^c=di,e^c=egh,f^c=fi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52574);