Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417493 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3822 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 3822 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,19,20,17)(16,18);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15)(14,17)(16,20)(18,19);(7,12)(9,11)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=eh,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238594, #72240616, #72262307, … |
| maximal normal subgroup of | #72238594, #72240616, #72262307, … |
| maximal quotient of | #72238595, #72258606, #72262391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356697);