Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,11,16,19)(9,12)(13,17,15,20)(14,18);(5,7)(6,10)(8,15,16,13)(9,14)(11,17,19,20)(12,18);(5,8)(6,11,12,19)(7,13)(9,16)(10,17,18,20)(14,15);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225628, #72225630, … |
| maximal normal subgroup of | #72225620, #72225628, #72225630, … |
| maximal quotient of | #72255677, #72265827 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305298);