Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
3048 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)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,22)(20,23)(26,29)(30,32);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,23,24,20)(25,26,28,29)(27,30,31,32);(2,6)(4,8)(11,15)(13,16)(18,22)(20,23)(25,27,28,31)(26,30,29,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=fg,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326572, #72326584, #72326623, … |
| maximal normal subgroup of | #72326572, #72326584, #72326623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293609);