Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485611 |
| 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 | C2×#534 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
901 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 901 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,32)(21,33)(23,29)(24,30)(25,38)(26,31)(27,34)(28,36)(35,40)(37,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,18,28)(12,20,23,35)(13,22,26,36)(16,27,29,39)(17,19,31,33)(24,34,25,37)(30,32,38,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,25)(14,23)(15,27)(17,30)(18,29)(19,32)(21,35)(22,37)(24,26)(28,39)(31,38)(33,40)(34,36);(2,7)(6,8)(11,21)(12,23)(15,28)(16,29)(19,33)(22,36)(24,25)(30,38)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dgi,e^b=eg,f^b=fhi,d^c=di,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, 392978);