Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle, prime radical | C23 | D4⋊C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
222 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(6,11)(7,15)(8,17)(9,16)(12,13)(14,20)(18,19)(21,22,24,30,26,31,33,29,27,32,23,28,25,35,34,36);(2,4)(5,6,8,12,9,13,7,11,10,14,15,19,16,18,17,20)(22,28,31,36,32,30,35,29)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fg,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72216536, #72225188, #72225193, … |
| maximal normal subgroup of | #72216536, #72225188, #72225193, … |
| maximal quotient of | #72216534, #72216535, #72227004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127);