Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486811 |
| 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 | #2619 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 746 conjugacy classes of subgroups — too many to draw. Only its 116 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)(17,18,23,28)(19,29,32,44)(20,25,21,26)(22,36,38,27)(24,37,40,31)(30,45,34,48)(33,43,35,41)(39,47,42,46);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,30)(21,34)(22,37)(23,32)(25,39)(26,42)(27,44)(28,40)(29,36)(31,38)(33,47)(35,46)(41,48)(43,45);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,23,21)(18,25,28,26)(19,30,32,34)(22,35,38,33)(24,39,40,42)(27,41,36,43)(29,45,44,48)(31,47,37,46);(3,10)(7,14)(11,16)(13,15)(17,21,23,20)(18,26,28,25)(19,30,32,34)(22,35,38,33)(24,39,40,42)(27,43,36,41)(29,45,44,48)(31,46,37,47)Polycyclic presentation:
2^9|a^2=g,c^2=i,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=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, 394141);