Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484606 |
| 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 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 817 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,16)(12,23)(13,18)(14,28)(15,20)(17,25)(19,29)(21,35)(22,34)(24,38)(26,37)(27,31)(30,36)(32,39)(33,40);(1,2,4,6)(3,8,7,5)(9,11,15,13)(10,16,20,18)(12,21,24,26)(14,22,29,27)(17,30,32,33)(19,34,28,31)(23,37,38,35)(25,36,39,40);(9,12)(10,17,20,32)(11,21)(13,26)(14,25)(15,24)(16,30,18,33)(19,23,28,38)(22,36)(27,40)(29,39)(31,35,34,37);(9,13,15,11)(10,18,20,16)(12,21,24,26)(14,27,29,22)(17,30,32,33)(19,34,28,31)(23,35,38,37)(25,36,39,40)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391923);