Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C2 | #11494 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,12,6,14,26,41,7,11,25,37,27,38,53,60,8,13,23,40,28,33,45,58,4,9,21,30,22,31,50,55)(3,15,17,42,18,43,47,62,19,35,46,39,48,57,63,61,20,29,24,51,49,54,52,64,16,10,32,34,44,36,56,59);(2,9,13,11)(5,23)(10,29,35,15)(12,30,40,37)(14,31,33,38)(17,24)(21,25)(26,45)(32,46)(34,51,39,42)(36,54,57,43)(41,55,58,60)(47,52)(50,53)(56,63)(59,64,61,62);(1,3,6,18,7,19,27,48,8,20,28,49,4,16,22,44)(2,10,33,57,11,15,31,54,13,35,14,36,9,29,38,43)(5,24,26,52,25,32,53,56,23,17,45,47,21,46,50,63)(12,39,58,59,37,51,55,62,40,34,41,61,30,42,60,64)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=fg,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,e^b=efi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287401, #72342968, #72342972, … |
| maximal normal subgroup of | #72287401, #72342968, #72342972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56935);