Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C23 | #448 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2)(3,9,11,16)(4,6)(5,8)(7,10,14,13)(12,15)(17,18,22,29,23,26,38,44,24,28,36,46,20,35,21,27)(19,30,34,43,31,45,32,40,33,47,37,25,39,41,48,42);(2,6)(5,12)(7,14)(10,13)(17,19,20,31,24,33,23,39)(18,25,35,42,28,43,26,40)(21,37,36,48,38,34,22,32)(27,45,46,47,44,41,29,30);(1,3,4,11)(2,7,6,14)(5,13,12,10)(8,16,15,9)(17,20,24,23)(18,26,28,35)(19,31,33,39)(21,22,38,36)(25,40,43,42)(27,46,44,29)(30,45,47,41)(32,34,48,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gi,e^2=hi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ehi,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44195);