Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3248 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,24)(19,23,21,20)(25,26,29,28)(27,31,32,30);(2,4)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,24,20);(3,8)(6,7)(10,13)(14,15)(18,24)(20,23)(25,27,29,32)(26,30,28,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,21,22,19)(18,20,24,23)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263168, #72263185, #72263190, … |
| maximal normal subgroup of | #72263168, #72263185, #72263190, … |
| maximal quotient of | #72318171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4883072);