Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485060 |
| 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 | #3497 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 780 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,16)(12,25)(13,18)(14,28)(15,20)(17,26)(19,30)(21,37)(22,32)(23,40)(24,34)(27,36)(29,33)(31,38)(35,39);(1,2)(3,6)(4,8)(5,7)(9,11,15,24)(10,16,20,34)(12,22,13,21)(14,29,30,23)(17,35,36,31)(18,37,25,32)(19,33,28,40)(26,38,27,39);(9,12,15,13)(10,17,20,36)(11,21,24,22)(14,27,30,26)(16,31,34,35)(18,28,25,19)(23,39,29,38)(32,40,37,33);(1,3,5,4)(2,6,7,8)(9,13,15,12)(10,18,20,25)(11,22,24,21)(14,27,30,26)(16,32,34,37)(17,28,36,19)(23,39,29,38)(31,40,35,33)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392430);