Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(11,15)(14,16)(17,18,22,23)(19,24,20,21)(26,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,15,16)(18,23)(19,20)(25,26)(27,32)(28,31)(29,30);(17,19,22,20)(18,21,23,24)(25,27,29,28)(26,31,30,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,21,23,24)(25,28,29,27)(26,31,30,32);(1,4)(2,7)(3,8)(5,6)(9,12,13,10)(11,16,15,14)(17,21)(18,20)(19,23)(22,24)(25,26)(27,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=h,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72329143, #72329146, #72329235, … |
| maximal normal subgroup of | #72329143, #72329146, #72329235, … |
| maximal quotient of | #72331034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4826981);