Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1333 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| socle | C2 | C2×#1333 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,13,8,15,24,42,9,16,7,32,4,11,23,14)(3,17,20,41,18,47,30,49,21,51,54,63,34,50,33,62)(5,25,27,45,28,46,53,40,29,59,31,12,22,43,58,44)(10,26,36,19,35,55,48,52,39,60,64,57,38,56,37,61);(1,3)(2,10)(4,18)(5,26)(6,30)(7,33)(8,34)(9,21)(11,35)(12,41)(13,37)(14,36)(15,38)(16,39)(17,43)(19,53)(20,24)(22,55)(23,54)(25,50)(27,52)(28,56)(29,60)(31,61)(32,48)(40,49)(42,64)(44,62)(45,63)(46,51)(47,59)(57,58);(1,4,9,8)(2,11,16,15)(3,18,21,34)(5,22,29,28)(6,23,7,24)(10,35,39,38)(12,40,45,44)(13,14,32,42)(17,47,51,50)(19,52,57,61)(20,30,54,33)(25,43,59,46)(26,55,60,56)(27,58,31,53)(36,48,64,37)(41,49,63,62)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72223646, #72223653, #72223683, … |
| maximal normal subgroup of | #72223646, #72223653, #72223683, … |
| maximal quotient of | C4×#7390, C2×#150074, #72286494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57270);