Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
4032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,23,24,21)(25,26,28,29)(27,31,32,30);(3,8)(5,7)(10,13)(11,15)(26,29)(30,31);(9,10,12,13)(11,14,15,16)(17,19,20,24)(18,21,22,23)(25,27)(26,30)(28,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=i,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72336292, #72336316, #72336477, … |
| maximal normal subgroup of | #72336292, #72336316, #72336477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5894887);