Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #99 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11465 |
| prime radical | #532 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,6,13,7,15,30,44,8,12,29,40,32,41,55,61,9,14,31,43,33,45,56,63,5,26,17,42,28,54,48,62)(3,16,19,46,20,47,49,64,18,10,4,11,22,35,23,39,21,36,24,37,50,58,52,59,27,34,25,38,51,57,53,60);(1,3)(2,10)(4,17)(5,27)(6,24)(7,20)(8,18)(9,21)(11,13)(12,36)(14,34)(15,35)(16,26)(19,31)(22,32)(23,48)(25,29)(28,51)(30,52)(33,50)(37,40)(38,43)(39,44)(41,58)(42,46)(45,57)(47,54)(49,56)(53,55)(59,61)(60,63)(62,64);(1,4,7,23,8,24,32,52,9,25,33,53,5,19,28,49)(2,11,15,39,12,37,41,59,14,38,45,60,26,46,54,64)(3,17,20,48,18,6,22,30,21,29,50,55,27,31,51,56)(10,13,35,44,36,40,58,61,34,43,57,63,16,42,47,62)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,d^b=di,e^b=efi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287401, #72287798, #72287804, … |
| maximal normal subgroup of | #72287401, #72287798, #72287804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56867);