Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×H(5,𝔽2) | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(2,3)(5,6,9,13)(7,15,17,11)(8,12,20,28)(10,14,22,30)(16,25,34,33)(18,31,23,26)(19,36,35,27)(21,32,24,29);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,19,20,35)(10,23,22,18)(12,27,28,36)(14,31,30,26)(16,24,34,21)(25,32,33,29);(5,8)(6,12)(7,16)(9,20)(10,24)(11,25)(13,28)(14,32)(15,33)(17,34)(18,35)(19,23)(21,22)(26,36)(27,31)(29,30);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407973);