Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622953 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#459 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,9)(5,7)(10,18)(11,12)(13,22)(14,26)(15,23)(17,19)(20,21)(24,25)(27,31)(28,32)(29,30);(1,2,7,11,8,12,5,6)(3,9,21,26,15,23,14,20)(4,10,19,25,16,24,17,18)(13,22,30,32,27,31,28,29);(1,3)(2,9)(4,13)(5,14)(6,20)(7,21)(8,15)(10,22)(11,26)(12,23)(16,27)(17,28)(18,29)(19,30)(24,31)(25,32);(1,4)(2,10)(3,13)(5,17)(6,18)(7,19)(8,16)(9,22)(11,25)(12,24)(14,28)(15,27)(20,29)(21,30)(23,31)(26,32);(4,16)(10,24)(13,27)(17,19)(18,25)(22,31)(28,30)(29,32)Polycyclic presentation:
2^9|b^2=fi,f^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215805, #72215816, #72215833, … |
| maximal normal subgroup of | #72215805, #72215816, #72215833, … |
| maximal quotient of | C2×#7622954, #72226792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530150);