Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1909 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1909 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 36 points:
(5,6)(7,17)(8,14)(9,12)(10,24)(11,16)(13,20)(15,26)(18,34)(19,33)(21,32)(22,28)(23,31)(25,30)(27,36)(29,35);(1,2)(3,4)(6,12)(8,9)(13,29)(14,16)(15,31)(17,33)(18,19)(22,23)(24,28)(27,30)(32,34)(35,36);(2,4)(5,7,11,21)(6,13,16,30)(8,18,9,19)(10,25,26,20)(12,27,14,29)(15,32,24,17)(22,36,23,35)(28,34,31,33);(5,8)(7,18)(9,11)(10,23)(12,14)(19,21)(20,36)(22,26)(25,35)(27,29)(28,31)(33,34)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72210404, #72266866, #72267264, … |
| maximal normal subgroup of | #72210404, #72266866, #72267264, … |
| maximal quotient of | #72210355, #72266734, #72266752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407065);