Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152861 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13168 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #510 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
167 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,9)(3,14,18,43)(4,7)(5,24)(6,28)(10,32,35,40)(11,33)(12,39)(13,21)(15,34,47,56)(16,22)(17,45,20,41)(19,50,53,63)(23,29)(25,49)(26,52)(27,51)(30,38)(36,61,60,59)(42,64,54,44)(46,55)(48,57,58,62);(1,2)(3,15,18,47)(4,21)(5,25)(6,12)(7,9)(8,13)(10,20,35,17)(11,27)(14,40,43,32)(16,49)(19,48,53,58)(22,30)(23,37)(24,46)(26,55)(28,31)(29,39)(33,52)(34,41,56,45)(36,54,60,42)(38,51)(44,62,64,57)(50,59,63,61);(1,3,16,50,6,19,5,14,4,17,24,44,23,42,22,41,8,18,52,63,29,53,27,43,7,20,51,64,28,54,26,45)(2,10,33,57,12,36,11,34,21,47,38,59,37,58,55,40,13,35,49,62,39,60,25,56,9,15,46,61,31,48,30,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=egh,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225556, #72225573, #72225574, … |
| maximal normal subgroup of | #72225556, #72225573, #72225574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60057);