Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419459 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
2085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,15,10)(8,11,16,19)(9,12)(14,17)(18,20);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,20)(17,19);(2,4)(5,8)(6,11,12,19)(7,14)(9,16)(10,17,13,18)(15,20);(10,13)(17,18)Polycyclic presentation:
2^9|a^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225628, #72225633, … |
| maximal normal subgroup of | #72225620, #72225628, #72225633, … |
| maximal quotient of | #72251435, #72251445, #72265796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326655);