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 | |
3170 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)(9,10)(11,13)(12,15)(14,16)(18,21)(22,23)(26,29)(27,31);(2,5)(6,8)(17,18,20,21)(19,23,24,22)(25,26,28,29)(27,32,31,30);(10,11)(14,15)(18,21)(22,23)(27,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=f,d^2=g,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 | #72326573, #72326585, #72326623, … |
| maximal normal subgroup of | #72326573, #72326585, #72326623, … |
| maximal quotient of | #72328994 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863953);