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 | #44872 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 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)(9,10)(11,13)(12,14)(15,16)(19,25)(22,27)(26,31)(29,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,18)(19,27)(20,24)(21,23)(22,25)(26,32)(28,30)(29,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,20,26,21,25,28,31)(18,22,24,29,23,27,30,32);(2,6)(7,8)(18,23)(22,27)(24,30)(29,32)Polycyclic presentation:
2^9|c^2=f,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 | #72257553, #72257568, #72264324 |
| maximal normal subgroup of | #72257553, #72257568, #72264324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517465);