Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
734 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 82 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)(19,27)(20,31,34,24)(21,35,36,25)(22,26)(23,29,38,45)(28,43,32,41)(30,42)(33,40)(37,44)(39,47)(46,48);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,22,30)(18,23,26,38)(20,32,34,28)(21,29,36,45)(24,39,31,37)(25,27,35,42)(33,47,48,44)(40,43,46,41);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(17,20,22,34)(18,24,26,31)(19,28,30,32)(21,33,36,48)(23,37,38,39)(25,40,35,46)(27,41,42,43)(29,44,45,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fhi,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40294);