Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×#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 ◁ C8 ◁ C2×C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 243 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:
(1,2,6,13,8,14,31,46,9,15,23,45,33,47,53,64,7,16,30,34,32,48,57,58,5,12,18,43,29,44,50,63)(3,17,20,38,22,49,51,60,19,37,4,11,24,39,26,40,21,42,27,41,52,61,56,62,28,10,25,35,54,36,55,59);(1,3)(2,10)(4,23)(5,28)(6,20)(7,21)(8,22)(9,19)(11,13)(12,42)(14,36)(15,17)(16,37)(18,25)(24,33)(26,53)(27,30)(29,54)(31,51)(32,52)(34,35)(38,43)(39,48)(40,46)(41,45)(44,61)(47,49)(50,55)(56,57)(58,59)(60,63)(62,64);(1,4,8,26,9,27,33,56,7,25,32,55,5,20,29,51)(2,11,14,40,15,41,47,62,16,35,48,59,12,38,44,60)(3,18,22,50,19,6,24,31,21,23,52,53,28,30,54,57)(10,34,36,58,17,43,49,63,37,13,39,46,42,45,61,64)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,c^b=ci,d^b=di,e^b=efi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287376, #72287804, #72287807, … |
| maximal normal subgroup of | #72287376, #72287804, #72287807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56868);