Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(18,22)(19,23)(25,26)(27,30)(28,31)(29,32);(2,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,24,23,20)(25,27)(26,29)(28,32)(30,31);(3,7)(6,8)(10,14)(12,15)(18,22)(20,24)(25,27,28,32)(26,30,31,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72328082, #72328114, #72328181, … |
| maximal normal subgroup of | #72328082, #72328114, #72328181, … |
| maximal quotient of | #72328507, #72331388 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2163024);