Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3130 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(17,18,21,23)(19,20,24,22)(25,26)(27,30)(28,29)(31,32);(2,5)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,23,20)(26,29)(30,32);(3,8)(6,7)(10,15)(11,16)(18,23)(20,22)(25,27,28,31)(26,30,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72334163, #72334165, #72334201, … |
| maximal normal subgroup of | #72334163, #72334165, #72334201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4236317);