Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143438 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,9)(4,13)(8,14)(10,15)(11,12)(17,18,23,28)(19,29,34,47)(20,35,38,45)(21,25,22,27)(24,39,42,31)(26,37,44,36)(30,40,48,41)(32,43,33,46);(1,2,5,8)(3,10,12,15)(4,14,13,6)(7,11,16,9)(17,19,23,34)(18,24,28,42)(20,30,38,48)(21,33,22,32)(25,40,27,41)(26,43,44,46)(29,45,47,35)(31,37,39,36);(1,3)(2,7)(4,11)(5,12)(6,15)(8,16)(9,13)(10,14)(18,25)(19,30)(20,36)(21,22)(24,29)(26,44)(27,28)(31,33)(32,39)(34,48)(37,38)(40,43)(41,46)(42,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50656);