Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
3360 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)(9,10,13,14)(11,15,16,12)(17,18,20,21)(19,22,24,23)(25,26)(27,31)(28,32)(29,30);(3,8)(5,7)(10,14)(11,16)(17,18,20,21)(19,23,24,22)(26,30)(27,32);(11,16)(12,15)(18,21)(22,23)(25,27,29,32)(26,31,30,28);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,15)(10,11,14,16)(17,19,20,24)(18,22,21,23)(25,28,29,31)(26,27,30,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #72300700, #72300750, #72300775, … |
| maximal normal subgroup of | #72300700, #72300750, #72300775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6075000);