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 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
815 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 815 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10)(11,15)(12,22)(13,26)(14,18)(16,25)(17,28)(19,36)(20,34)(21,31)(23,40)(24,35)(27,33)(29,37)(30,39)(32,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,31)(12,23,24,19)(13,20,28,39)(16,32,33,29)(17,34,26,30)(22,36,35,40)(25,38,27,37);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,35)(18,33)(20,38)(21,23)(22,26)(25,28)(30,40)(31,32)(34,36)(37,39);(2,7)(6,8)(12,24)(16,33)(19,23)(22,35)(25,27)(29,32)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=i,c^2=g,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,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, 393018);