Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484162 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,22,26)(19,27,30,42)(20,24,32,40)(21,34,36,25)(23,35,38,29)(28,43,31,45)(33,48,47,41)(37,46,39,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,38)(20,31,32,28)(21,35,36,29)(24,39,40,37)(25,42,34,27)(33,44,47,46)(41,43,48,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,32)(21,36)(24,40)(28,31)(29,35)(37,39)(41,48)(43,45);(17,20,22,32)(18,24,26,40)(19,28,30,31)(21,33,36,47)(23,37,38,39)(25,41,34,48)(27,43,42,45)(29,44,35,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391628);