Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3796 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)(3,6)(4,8)(5,7)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,23,24,21)(25,26,28,30)(27,29,31,32);(1,3,5,8)(2,6,7,4)(10,13)(14,16)(19,24)(21,23)(25,27,28,31)(26,29,30,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,29)(28,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263269, #72263290, #72263304, … |
| maximal normal subgroup of | #72263269, #72263290, #72263304, … |
| maximal quotient of | #72328511 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2049360);