Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| 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 | C2×#585 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2105 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 2105 conjugacy classes of subgroups — too many to draw. Only its 194 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:
(1,2)(3,4)(5,6,12,8)(7,17,21,30)(9,16,11,14)(10,23,26,15)(13,25,22,20)(18,33,31,36)(19,35,27,34)(24,29,28,32);(5,7)(6,13)(8,22)(9,19)(10,25)(11,27)(12,21)(14,18)(15,30)(16,31)(17,23)(20,26)(24,36)(28,33)(29,35)(32,34);(2,4)(5,8,12,6)(7,18,21,31)(9,14,11,16)(10,23,26,15)(13,27,22,19)(17,33,30,36)(20,35,25,34)(24,29,28,32);(5,9,12,11)(6,14,8,16)(7,19,21,27)(10,24,26,28)(13,18,22,31)(15,29,23,32)(17,34,30,35)(20,33,25,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406107);