Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| 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 | |
531 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 531 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)(8,19)(9,22)(11,25)(12,20)(13,24)(16,21)(17,33)(18,32)(23,29)(26,30)(27,36)(28,35)(31,34);(1,2)(3,4)(5,6,10,14)(7,11,17,27)(8,20,23,34)(9,24,22,13)(12,29,31,19)(15,30,33,35)(16,28,18,26)(21,25,32,36);(2,4)(5,7)(6,11)(8,21)(9,16)(10,17)(12,30)(13,26)(14,27)(15,19)(18,22)(20,25)(23,32)(24,28)(29,33)(31,35)(34,36)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72217759 |
| maximal normal subgroup of | #72217759 |
| maximal quotient of | #72216706, #72216712, #72216714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50393);