Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | #1852 |
| prime radical | C2×#41 | Q16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #2965 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 255 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(17,18,21,26,23,27,22,37)(19,28,31,45,30,44,38,42)(20,32,35,43,34,47,36,25)(24,33,40,46,39,48,41,29);(1,3,6,13,7,15,5,14)(2,8,9,16,4,12,10,11)(17,19,21,31,23,30,22,38)(18,24,26,40,27,39,37,41)(20,33,35,46,34,48,36,29)(25,42,32,28,43,45,47,44);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(20,34)(25,43)(28,44)(29,46)(32,47)(33,48)(35,36)(42,45)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=ehi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43654);