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 | |
3539 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,22)(21,23)(26,30)(28,32);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,22)(23,24)(26,30)(27,31);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,13,14,15)(17,20,21,24)(18,22,23,19)(25,26)(27,28)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28)(29,31)(30,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|b^2=fi,c^2=f,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 | #72329105, #72329120, #72329204 |
| maximal normal subgroup of | #72329105, #72329120, #72329204 |
| maximal quotient of | #72330554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4521954);