Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485657 |
| 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 | #12100 | C2 |
| 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 | |
868 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 868 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,18)(14,28)(15,20)(17,26)(19,30)(21,32)(22,38)(23,34)(25,40)(27,36)(29,33)(31,37)(35,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,25)(13,21)(14,29)(15,23)(17,35)(18,32)(19,33)(20,34)(22,30)(24,40)(26,37)(27,39)(28,38)(31,36);(9,12,15,13)(10,17,20,36)(11,21,23,25)(14,27,30,26)(16,31,34,35)(18,28,24,19)(22,39,29,37)(32,33,40,38);(1,3,5,4)(2,7,8,6)(9,13,15,12)(10,18,20,24)(11,21,23,25)(14,27,30,26)(16,32,34,40)(17,28,36,19)(22,37,29,39)(31,33,35,38)Polycyclic presentation:
2^9|c^2=i,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,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, 393118);