Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132559 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 559 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:
(1,2,5,9,6,10,4,14)(3,11,12,7,13,8,15,16)(17,18)(19,28,31,42)(20,32,35,43)(21,26,37,25)(22,27)(23,30)(24,34)(29,38)(33,40)(36,39)(41,48)(44,47,46,45);(1,3)(2,7)(4,15)(5,12)(6,13)(8,9)(10,16)(11,14)(17,19,22,31)(18,23,27,39)(20,33,35,29)(21,36,37,30)(24,40,41,38)(25,42,26,28)(32,46,43,44)(34,45,48,47);(1,3)(2,8)(4,12)(5,15)(6,13)(7,9)(10,11)(14,16)(17,20,22,35)(18,24,27,41)(19,29,31,33)(21,34,37,48)(23,38,39,40)(25,43,26,32)(28,44,42,46)(30,45,36,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fgi,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, 40128);