Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,20,24)(25,26)(27,29)(28,30)(31,32);(2,4)(3,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,20)(18,21,23,24)(27,31)(29,32);(2,4)(6,7)(10,13)(14,15)(18,23)(19,20)(25,26,28,30)(27,29,31,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,21,23,24)(25,26)(27,29)(28,30)(31,32);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,21)(18,19)(20,23)(22,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fgh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263336, #72263337 |
| maximal normal subgroup of | #72263336, #72263337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5025167);