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) = C25×C4 |
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 | |
2905 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)(11,16,12,15)(17,18)(19,22)(20,24)(21,23)(25,26,28,31)(27,30,32,29);(3,8)(5,7)(10,14)(15,16)(17,19,21,24)(18,20,23,22)(25,27,28,32)(26,29,31,30);(9,11,13,12)(10,15,14,16)(17,19,21,24)(18,22,23,20)(25,26,28,31)(27,30,32,29);(1,3)(2,5)(4,8)(6,7)(9,12,13,11)(10,16,14,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=f,d^2=g,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 | #72300713, #72300740, #72300760, … |
| maximal normal subgroup of | #72300713, #72300740, #72300760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5896014);