Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8843 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,11,7,14,31,40,8,15,32,41,30,13,5,12)(3,16,18,9,21,34,52,35,22,46,53,36,20,55,19,43)(4,23,26,38,28,37,60,39,29,58,24,44,27,42,25,10)(17,47,49,45,50,33,48,57,51,62,59,64,56,63,54,61);(1,3,6,18,7,21,31,52,8,22,32,53,30,20,5,19)(2,9,11,34,14,35,40,46,15,36,41,55,13,43,12,16)(4,17,26,49,28,50,60,48,29,51,24,59,27,56,25,54)(10,33,23,57,38,62,37,64,39,63,58,61,44,47,42,45);(1,4)(2,10)(3,17)(5,25)(6,26)(7,28)(8,29)(9,33)(11,23)(12,42)(13,44)(14,38)(15,39)(16,45)(18,49)(19,54)(20,56)(21,50)(22,51)(24,32)(27,30)(31,60)(34,57)(35,62)(36,63)(37,40)(41,58)(43,47)(46,64)(48,52)(53,59)(55,61)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=egi,g^b=gi,d^c=dgi,e^c=egi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43089);