Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487643 |
| 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 | #3496 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
795 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 795 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:
(2,7)(4,5)(8,14)(9,13)(10,12)(15,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,30)(26,36)(28,42)(29,41)(31,46)(32,39)(34,38)(35,40)(37,44)(43,48)(45,47);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,16)(12,14)(17,19,22,29)(18,23,25,38)(20,28,32,43)(21,34,36,30)(24,37,39,47)(26,41,33,27)(31,45,35,44)(40,48,46,42);(1,3,6,11)(2,8,9,16)(4,12,5,10)(7,14,13,15)(17,20,22,32)(18,24,25,39)(19,28,29,43)(21,35,36,31)(23,37,38,47)(26,40,33,46)(27,42,41,48)(30,45,34,44);(1,4,6,5)(2,7,9,13)(3,10,11,12)(8,15,16,14)(18,25)(19,29)(24,39)(26,33)(28,43)(30,34)(40,46)(44,45)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,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, 395428);