Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486859 |
| 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×C4⋊C4 | C24 |
| prime radical | #18324 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
822 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 822 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,30)(21,33)(22,26)(24,32)(25,35)(27,42)(28,40)(29,38)(31,41)(34,39)(36,44)(37,46)(43,47)(45,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,29)(18,23,26,38)(20,27,31,43)(21,28,35,46)(24,36,39,48)(25,40,33,37)(30,47,41,42)(32,44,34,45);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,34,35,32)(23,36,38,48)(25,41,33,30)(28,45,46,44)(37,47,40,42);(3,10)(7,14)(11,16)(13,15)(20,31)(24,39)(27,43)(30,41)(32,34)(36,48)(42,47)(44,45)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 394058);