Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131844 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8443 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#3293 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,6,12,7,30,29,11,5,13,25,47,26,10,39,42)(3,14,17,45,18,8,32,35,16,40,55,34,31,36,27,43)(4,19,22,37,20,57,21,38,24,46,28,41,23,44,58,9)(15,49,52,33,50,60,51,61,54,64,56,62,53,63,59,48);(1,3,5,16)(2,8,13,36)(4,15,24,54)(6,27,25,32)(7,31,26,18)(9,33,38,62)(10,40,30,14)(11,43,42,35)(12,45,47,34)(17,29,55,39)(19,49,46,64)(20,53,23,50)(21,56,58,52)(22,59,28,51)(37,48,41,61)(44,60,57,63);(1,4)(2,9)(3,15)(5,24)(6,28)(7,20)(8,33)(10,41)(11,44)(12,46)(13,38)(14,48)(16,54)(17,56)(18,50)(19,47)(21,39)(22,25)(23,26)(27,51)(29,58)(30,37)(31,53)(32,59)(34,49)(35,63)(36,62)(40,61)(42,57)(43,60)(45,64)(52,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=g,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dh,e^b=egi,f^b=fghi,g^b=gi,d^c=dgi,e^c=egi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39041);