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 | #28504 | 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 | |
537 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 537 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:
(5,6,10,14)(7,15,18,34)(8,20,24,36)(9,25,23,13)(11,26,28,17)(12,30,32,21)(16,31,19,33)(22,27,35,29);(1,2)(3,4)(8,21)(9,23)(12,20)(13,25)(15,31)(16,19)(17,22)(24,30)(26,35)(27,29)(32,36)(33,34);(2,4)(5,7)(6,11)(8,22)(9,16)(10,18)(12,31)(13,27)(14,28)(15,20)(17,21)(19,23)(24,35)(25,29)(26,30)(32,33)(34,36)Polycyclic presentation:
2^9|a^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 | #72226861, #72226869, #72226872, … |
| maximal normal subgroup of | #72226861, #72226869, #72226872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50394);