Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
10322 conjugacy classes of subgroups; only the 1042 normal subgroups are shown.
This group has 10322 conjugacy classes of subgroups — too many to draw. Only its 1042 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,8)(9,10,12,11)(14,17)(15,19);(2,4)(6,8)(10,11)(14,17)(18,20);(1,2)(3,4)(5,6,7,8)(9,11,12,10);(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=f,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240155, #72240157, #72240173, … |
| maximal normal subgroup of | #72240155, #72240157, #72240173, … |
| maximal quotient of | #72317079, #72323527, #72323599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862527);