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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3188 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)(11,12)(13,14)(15,16)(17,18,20,21)(19,23,24,22)(25,26,28,30)(27,31,32,29);(2,6)(3,7)(9,10,13,14)(11,15,16,12)(18,21)(22,23)(25,27)(26,29)(28,32)(30,31);(2,6)(4,8)(10,14)(12,15)(17,19,20,24)(18,22,21,23)(25,26,28,30)(27,29,32,31);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,32)(30,31);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fh,b^2=h,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 | #72329068, #72329131, #72329177 |
| maximal normal subgroup of | #72329068, #72329131, #72329177 |
| maximal quotient of | #72329903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4540029);