Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×C4×D16 |
| prime radical | #33981 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 176 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:
(1,2,6,13,8,11,29,14)(3,15,18,46,20,45,19,51)(4,10,22,38,23,37,7,30)(5,24,26,55,28,54,60,56)(9,32,35,50,34,49,43,17)(12,21,41,52,39,53,42,27)(16,44,47,63,48,64,31,40)(25,33,57,62,59,61,58,36);(1,3)(2,9)(4,16)(5,25)(6,18)(7,31)(8,20)(10,33)(11,34)(12,40)(13,35)(14,43)(15,24)(17,27)(19,29)(21,32)(22,47)(23,48)(26,57)(28,59)(30,36)(37,61)(38,62)(39,63)(41,44)(42,64)(45,54)(46,55)(49,53)(50,52)(51,56)(58,60);(1,4,6,22,8,23,29,7)(2,10,13,38,11,37,14,30)(3,16,18,47,20,48,19,31)(5,21,26,52,28,53,60,27)(9,33,35,62,34,61,43,36)(12,24,41,55,39,54,42,56)(15,44,46,63,45,64,51,40)(17,25,32,57,50,59,49,58);(2,11)(9,34)(10,37)(12,39)(13,14)(15,45)(24,54)(30,38)(33,61)(35,43)(36,62)(40,63)(41,42)(44,64)(46,51)(55,56)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72223679, C2×#512773, #72225308, … |
| maximal normal subgroup of | #72223679, C2×#512773, #72225308, … |
| maximal quotient of | C8×#2263, C2×#512773 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419969);