Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488877 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61105 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 64 points:
(1,2,8,14)(3,9,18,34)(4,19,23,54)(5,13,7,11)(6,25,29,41)(10,35,38,22)(12,28,42,24)(15,44,47,64)(16,43,30,32)(17,49,51,39)(20,40,21,53)(26,57,59,33)(27,36,55,37)(31,58,61,46)(45,52,48,63)(50,62,56,60);(1,3)(2,9)(4,15)(5,16)(6,26)(7,30)(8,18)(10,31)(11,32)(12,39)(13,43)(14,34)(17,28)(19,52)(20,45)(21,48)(22,56)(23,47)(24,51)(25,57)(27,58)(29,59)(33,41)(35,50)(36,60)(37,62)(38,61)(40,64)(42,49)(44,53)(46,55)(54,63);(1,4)(2,10)(3,15)(5,20)(6,27)(7,21)(8,23)(9,31)(11,36)(12,40)(13,37)(14,38)(16,45)(17,50)(18,47)(19,25)(22,24)(26,58)(28,35)(29,55)(30,48)(32,60)(33,63)(34,61)(39,64)(41,54)(42,53)(43,62)(44,49)(46,59)(51,56)(52,57);(1,5,8,7)(2,11,14,13)(3,16,18,30)(4,20,23,21)(6,24,29,28)(9,32,34,43)(10,36,38,37)(12,25,42,41)(15,45,47,48)(17,26,51,59)(19,53,54,40)(22,55,35,27)(31,60,61,62)(33,39,57,49)(44,63,64,52)(46,50,58,56)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283166, #72283226, #72283274, … |
| maximal normal subgroup of | #72283166, #72283226, #72283274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396075);