Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132529 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2543 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
544 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 48 points:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,28)(20,32)(21,35,22,26)(23,27)(24,30,39,36)(25,33,40,37)(29,38)(31,43)(34,41)(42,45,47,44)(46,48);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,38)(26,28)(27,39)(32,47)(33,44)(34,46)(35,43)(37,45)(40,48)(41,42);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20,23,34)(18,25,27,40)(19,29,31,46)(21,33,22,37)(24,38,39,48)(26,41,35,32)(28,42,43,47)(30,44,36,45)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39806);