Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485649 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,27)(19,34)(20,32)(21,30)(23,33)(26,31)(28,36)(29,38)(35,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,14,21)(10,15,18,30)(12,19,23,35)(13,20,27,38)(16,28,31,40)(17,32,25,29)(22,39,33,34)(24,36,26,37);(9,12,14,23)(10,16,18,31)(11,19,21,35)(13,26,27,24)(15,28,30,40)(17,33,25,22)(20,37,38,36)(29,39,32,34);(2,6)(5,8)(12,23)(16,31)(19,35)(22,33)(24,26)(28,40)(34,39)(36,37)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392848);