Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #44869 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,25)(22,27)(26,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32);(10,14)(11,15)(17,18,21,23)(19,27,25,22)(20,24,28,30)(26,32,31,29);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,19,20,26,21,25,28,31)(18,22,24,29,23,27,30,32);(2,7)(6,8)(18,23)(22,27)(24,30)(29,32)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281486, #72281985, #72282683 |
| maximal normal subgroup of | #72281486, #72281985, #72282683 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7516973);