Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
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 | |
3735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(10,14)(11,15)(18,21)(22,24)(26,29)(30,31);(1,2)(3,8)(4,5)(6,7)(9,10,13,14)(11,12,15,16)(19,23)(22,24)(25,26)(27,30)(28,29)(31,32);(3,7)(6,8)(10,14)(12,16)(17,18,20,21)(19,22,23,24)(25,26,28,29)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|b^2=fghi,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232578, #72232579, #72232675, … |
| maximal normal subgroup of | #72232578, #72232579, #72232675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389632);