Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137493 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #466 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 313 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6,8,11)(3,9,10,12)(17,18)(19,27)(20,32)(21,36)(22,38)(23,28)(24,30)(25,29)(26,37)(31,45)(33,44)(34,48)(35,40)(39,41)(42,46)(43,47);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,19,21,30,23,31,37,22)(18,24,26,27,28,38,36,45)(20,29,33,46,35,47,34,39)(25,40,41,44,43,32,42,48);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,20)(18,25)(19,29)(21,33)(22,39)(23,35)(24,40)(26,41)(27,44)(28,43)(30,46)(31,47)(32,38)(34,37)(36,42)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44689);