Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486827 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
765 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 765 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) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,42)(30,43)(31,47)(32,39)(33,40)(34,48)(35,41)(36,44)(37,46)(38,45);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,30)(18,23,26,37)(20,31,32,28)(21,29,35,46)(24,38,39,36)(25,27,41,43)(33,45,34,44)(40,47,48,42);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,32)(18,24,26,39)(19,28,30,31)(21,34,35,33)(23,36,37,38)(25,40,41,48)(27,42,43,47)(29,45,46,44);(3,10)(7,14)(11,16)(13,15)(20,32)(24,39)(28,31)(33,34)(36,38)(40,48)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394154);