Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487643 |
| 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×C22⋊C4 | C24 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,25)(14,17)(15,22)(18,28)(20,34)(21,33)(23,38)(24,31)(26,30)(27,32)(29,36)(35,40)(37,39);(1,2,5,7)(3,6,4,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,30)(18,33)(19,25)(22,28)(23,37)(24,35)(27,36)(31,39)(32,40)(34,38);(3,4)(6,8)(9,12,14,24)(10,16,17,31)(11,20,21,35)(13,27,28,23)(15,29,30,39)(18,32,25,38)(19,34,33,40)(22,37,26,36);(1,3,5,4)(2,6,7,8)(10,17)(11,21)(16,31)(18,25)(20,35)(22,26)(32,38)(36,37)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395270);