Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347363 |
| 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 | |
3160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,7)(3,11)(4,14)(8,20)(9,15)(10,18)(12,24)(13,27)(16,29)(19,22)(21,31)(23,30)(25,26)(28,32);(1,2)(3,8)(4,15)(5,9)(6,10)(7,14)(11,22)(12,21)(13,28)(16,30)(17,23)(18,29)(19,26)(20,27)(24,32)(25,31);(3,12)(4,16)(7,18)(8,21)(11,24)(13,27)(14,29)(15,30)(19,31)(20,28)(22,32)(25,26);(1,3)(2,8)(4,13)(5,12)(6,11)(7,19)(9,21)(10,22)(14,26)(15,28)(16,25)(17,24)(18,20)(23,32)(27,29)(30,31);(2,9)(7,15)(8,21)(10,23)(18,30)(19,28)(20,31)(22,32)Polycyclic presentation:
2^9|f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237982, #72237987, #72237988, … |
| maximal normal subgroup of | #72237982, #72237987, #72237988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254559);