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 | |
245 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 245 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)(3,14,15,42,18,45,19,54)(4,9)(5,23,26,57)(6,28)(7,13)(8,16,31,49,33,51,30,46)(10,27,38,60)(11,21)(12,22)(17,36,29,61,48,63,53,44)(20,55,56,41)(24,40,58,43,59,62,50,32)(25,35)(34,47,39,64)(37,52);(1,3)(2,8)(4,15)(5,24)(6,29)(7,18)(9,30)(10,36)(11,40)(12,31)(13,33)(14,41)(16,47)(17,52)(19,22)(20,50)(21,48)(23,42)(25,53)(26,59)(27,49)(28,43)(32,35)(34,61)(37,62)(38,63)(39,44)(45,55)(46,60)(51,64)(54,57)(56,58);(1,4,7,22)(2,9,13,12)(3,15,18,19)(5,20,26,56)(6,21,25,52)(8,30,33,31)(10,34,38,39)(11,35,37,28)(14,42,45,54)(16,46,51,49)(17,29,48,53)(23,55,57,41)(24,50,59,58)(27,47,60,64)(32,62,43,40)(36,61,63,44)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fi,e^2=g,f^2=h,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, #72223669, #72223685, … |
| maximal normal subgroup of | #72223646, #72223669, #72223685, … |
| maximal quotient of | C4×#7389, C2×#150071, #72227202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57267);