Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150990 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11964 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | #33720 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
268 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,12,29,14,26,43,32,13,23,36,24,46,25,11,8,16,31,9,7,33,4,22,6,15,30,35,55,42,27,44)(3,10,18,34,53,39,17,47,50,38,57,58,49,63,56,62,21,41,28,45,20,54,52,61,19,40,48,59,64,60,51,37);(2,9)(4,23)(5,27)(6,32)(7,24)(10,34)(11,14)(12,16)(13,36)(15,35)(17,48)(18,51)(19,50)(20,49)(22,42)(25,31)(26,30)(28,56)(29,55)(33,44)(37,39)(38,59)(40,58)(41,45)(43,46)(47,60)(52,57)(53,64)(54,62)(61,63);(1,3)(2,10)(4,17)(5,28)(6,19)(7,20)(8,21)(9,34)(11,37)(12,45)(13,38)(14,39)(15,40)(16,41)(18,31)(22,47)(23,48)(24,49)(25,51)(26,52)(27,56)(29,53)(30,57)(32,50)(33,54)(35,58)(36,59)(42,60)(43,61)(44,62)(46,63)(55,64)Polycyclic presentation:
2^9|a^2=e,d^2=gh,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,e^c=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72286285, #72343285, #72343287, … |
| maximal normal subgroup of | #72286285, #72343285, #72343287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58186);