Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3349 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)(4,8)(5,6)(11,15)(14,16)(18,21)(19,23)(25,26)(27,28)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,15,16)(17,18)(19,24)(20,21)(22,23)(26,30)(27,31);(2,6)(3,8)(19,23)(22,24)(25,27,29,31)(26,28,30,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,31)(30,32);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,18)(19,24)(20,21)(22,23)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|b^2=fhi,c^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300937, #72327607, #72327661, … |
| maximal normal subgroup of | #72300937, #72327607, #72327661, … |
| maximal quotient of | #72328988, #72329163, #72331572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2708283);