Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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×D4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1903 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,13)(9,22)(10,25)(11,16)(12,19)(14,23)(15,27)(18,28)(20,36)(21,29)(24,33)(26,32)(30,35)(31,34);(1,2)(3,4)(5,7,11,8)(6,12,16,29)(9,18,24,20)(10,21,27,19)(13,25,17,15)(14,28,32,36)(22,31,33,30)(23,35,26,34);(2,4)(5,8,11,7)(6,13,16,17)(9,18,24,20)(10,21,27,19)(12,15,29,25)(14,31,32,30)(22,36,33,28)(23,34,26,35);(5,9,11,24)(6,14,16,32)(7,18,8,20)(10,26,27,23)(12,28,29,36)(13,30,17,31)(15,33,25,22)(19,35,21,34)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407038);