Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6730 conjugacy classes of subgroups; only the 1066 normal subgroups are shown.
This group has 6730 conjugacy classes of subgroups — too many to draw. Only its 1066 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,8)(10,12);(1,2,3,4)(13,14)(15,18)(16,17)(19,20);(5,6,7,8)(14,17)(18,20);(9,10,11,12);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch128 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2397, C2×#608453, #72280449, … |
| maximal normal subgroup of | D4×#2397, C2×#608453, #72280449, … |
| maximal quotient of | C2×#608453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515649);