Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3189 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)(3,5)(4,6)(7,8)(10,14)(11,15)(17,18)(19,20)(21,22)(23,24)(26,27)(28,32);(2,3)(6,7)(9,10,13,14)(11,16,15,12)(17,18,21,22)(19,20,23,24)(25,26,29,27)(28,30,32,31);(4,8)(6,7)(10,14)(12,16)(18,22)(20,24)(25,26,29,27)(28,31,32,30);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,19,21,23)(18,20,22,24)(25,27,29,26)(28,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,23)(19,22)(21,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|b^2=fghi,c^2=fgi,d^2=hi,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 | #72266259, #72266265, #72266277, … |
| maximal normal subgroup of | #72266259, #72266265, #72266277, … |
| maximal quotient of | #72318601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2342495);