Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143324 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| 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 | |
461 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 461 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(7,15,19,34)(8,20)(9,24)(11,26,28,36)(12,29)(13,25)(16,30,27,35)(17,33,18,22)(21,31)(23,32);(1,2)(3,4)(5,6,10,14)(7,16,19,27)(8,21,23,29)(9,25,24,13)(11,17,28,18)(12,20,31,32)(15,26,34,36)(22,35,33,30);(2,4)(5,7)(6,11)(8,22)(9,17)(10,19)(12,30)(13,27)(14,28)(15,32)(16,25)(18,24)(20,34)(21,26)(23,33)(29,36)(31,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217759 |
| maximal normal subgroup of | #72217759 |
| maximal quotient of | #72214389, #72214416, #72214428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50521);