Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,20,22,23)(26,29)(30,31);(3,7)(6,8)(10,15)(11,16)(18,22)(19,24)(25,27,28,32)(26,30,29,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|b^2=fgi,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232647, #72232685, #72232703, … |
| maximal normal subgroup of | #72232647, #72232685, #72232703, … |
| maximal quotient of | #72330303, #72330319 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4524318);