Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
3211 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,24,23,20)(25,26)(27,31)(28,29)(30,32);(2,4)(3,8)(11,16)(13,15)(25,27)(26,30)(28,31)(29,32);(3,8)(6,7)(18,22)(19,23)(27,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,23)(18,20,22,24)(25,27)(26,31)(28,30)(29,32);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,22)(21,24)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
| maximal subgroup of | #72263346, #72300938, #72300947, … |
| maximal normal subgroup of | #72263346, #72300938, #72300947, … |
| maximal quotient of | #72330831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5895936);