Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2932 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,24)(22,23)(25,26,28,30)(27,29,31,32);(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,21)(20,23)(22,24)(27,31)(29,32);(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,21)(18,19,23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21,22,20)(18,19,23,24)(25,27,28,31)(26,29,30,32);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=hi,d^2=gi,e^2=fhi,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 | #72236953, #72236962, #72262673, … |
| maximal normal subgroup of | #72236953, #72236962, #72262673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2507701);