Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18464 conjugacy classes of subgroups; only the 1138 normal subgroups are shown.
This group has 18464 conjugacy classes of subgroups — too many to draw. Only its 1138 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:
(2,4)(5,6,7,8)(10,12)(15,19)(17,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,16,18)(15,20,19,17);(5,6)(7,8)(13,15,16,19)(14,17,18,20);(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72210242, #72210333, #72210339, … |
| maximal normal subgroup of | #72210242, #72210333, #72210339, … |
| maximal quotient of | C2×#613102, #72267485, #72268214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 520298);