Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130254 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #62373 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,20,26,21,37,35,41,23,28,36,40,22,27,34,24)(19,29,32,47,38,46,48,43,33,45,30,39,31,25,42,44);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(18,24)(20,34)(21,22)(25,39)(26,27)(28,41)(29,43)(30,48)(31,38)(32,42)(35,36)(37,40)(44,45)(46,47);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,19,23,33)(18,25,28,46)(20,32,36,30)(21,38,22,31)(24,39,41,47)(26,44,40,43)(27,45,37,29)(34,42,35,48)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37660);