Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10)(11,13)(12,14)(15,16)(18,19)(20,24)(25,26,29,30)(27,28,31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,13)(14,16)(18,19)(22,23)(26,30)(27,31);(2,3)(6,7)(10,12)(11,16)(17,18,21,19)(20,22,24,23)(25,27,29,31)(26,28,30,32);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,13,16,15)(17,19,21,18)(20,23,24,22)(25,27)(26,28)(29,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,16)(11,12)(14,15)(17,20)(18,22)(19,23)(21,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=hi,c^2=fi,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72336356, #72336381, #72336429, … |
| maximal normal subgroup of | #72336356, #72336381, #72336429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2654417);