Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144059 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1677 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| 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 | |
260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 260 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,18)(7,19)(8,24)(9,27,28,16)(11,31,29,15)(12,32,30,17)(13,26)(14,25)(20,34)(21,35)(22,33)(23,36);(1,2)(3,4)(5,7)(6,13)(8,11)(9,21)(10,23)(12,20)(14,32)(15,33)(16,19)(17,35)(18,24)(22,30)(25,28)(26,29)(27,36)(31,34);(2,4)(5,8,9,22,12,25,29,23,10,26,28,20,30,21,11,7)(6,14,15,34,17,24,27,36,18,35,31,33,32,13,16,19)Polycyclic presentation:
2^9|a^2=h,c^2=d,d^2=g,e^2=hi,f^2=gi,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=dgi,e^b=egi,f^b=fg,g^b=gi,h^b=hi,e^c=eghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52638);