Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,15)(17,18,21,23)(19,20,24,22)(25,26,29,30)(27,31,32,28);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,21,24)(18,22,23,20)(26,30)(28,31);(3,7)(6,8)(10,14)(12,16)(18,23)(19,24)(25,27,29,32)(26,31,30,28);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,28)(26,27)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303399, #72303510, #72303515, … |
| maximal normal subgroup of | #72303399, #72303510, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3306574);