Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,18)(6,11)(7,13)(10,20)(12,28)(14,27)(15,24)(16,31)(17,26)(19,22)(21,30)(23,29)(25,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,18)(10,27)(11,22)(13,20)(15,17)(16,30)(21,26)(24,25)(28,31)(29,32);(1,4,6,17)(2,8,11,26)(3,12,14,15)(5,16,20,32)(7,21,22,23)(9,24,27,28)(10,25,18,31)(13,29,19,30);(1,3)(2,9)(4,15)(5,13)(6,14)(7,18)(8,24)(10,22)(11,27)(12,17)(16,30)(19,20)(21,31)(23,25)(26,28)(29,32);(5,20)(7,22)(10,18)(13,19)(16,32)(21,23)(25,31)(29,30)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dhi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233777, #72240648, #72249649, … |
| maximal normal subgroup of | #72233777, #72240648, #72249649, … |
| maximal quotient of | #72255717, #72255865, #72256149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407058);