Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
3567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,22)(21,23)(25,26,29,31)(27,28,32,30);(2,6)(3,7)(10,13)(14,15)(17,19)(18,20)(21,22)(23,24)(25,27)(26,30)(28,31)(29,32);(2,6)(4,8)(9,11,12,16)(10,14,13,15)(17,20,21,24)(18,22,23,19)(25,27,29,32)(26,30,31,28);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=g,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72328593, #72328607, #72328682 |
| maximal normal subgroup of | #72328593, #72328607, #72328682 |
| maximal quotient of | #72330554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4538492);