Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1578 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);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,25)(19,26,27,32)(20,23,28,24)(22,30,31,29);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(19,27)(22,31)(26,32)(29,30);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,28,30,21,27,20,29)(18,22,23,32,25,31,24,26);(2,6)(7,8)(17,20,21,28)(18,23,25,24)(19,29,27,30)(22,32,31,26)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257494, #72257516, #72264275 |
| maximal normal subgroup of | #72257494, #72257516, #72264275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517156);