Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394273 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,11)(10,12)(13,20)(14,18)(15,16)(17,19);(1,2,4,3)(5,7)(6,9)(8,15,17,13)(10,14)(11,16,19,20)(12,18);(5,8)(6,11)(7,13)(9,16)(10,17)(12,19)(14,15)(18,20);(1,3,4,2)(5,9)(6,7)(8,16)(10,18)(11,13)(12,14)(15,19)(17,20)Polycyclic presentation:
2^9|b^2=h,d^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225630, #72225640, #72225642, … |
| maximal normal subgroup of | #72225630, #72225640, #72225642, … |
| maximal quotient of | #72251460, #72265829, #72265831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312802);