Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347491 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3537 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)(3,12)(4,16)(5,9)(6,11)(7,14)(8,20)(10,19)(13,25)(15,27)(17,23)(18,24)(21,31)(22,30)(26,32)(28,29);(1,3)(2,7)(4,17)(5,13)(6,15)(8,21)(9,23)(10,25)(11,22)(12,16)(14,19)(18,31)(20,29)(24,27)(26,28)(30,32);(4,18)(10,26)(12,27)(14,30)(16,24)(17,31)(19,32)(25,28);(2,8)(7,21)(9,11)(10,24)(12,28)(16,26)(22,23)(25,27);(2,9)(3,13)(8,11)(10,16)(14,17)(15,29)(24,26)(30,31)Polycyclic presentation:
2^9|f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284955 |
| maximal normal subgroup of | #72284955 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254687);