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 | |
1569 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)(19,26)(23,31)(27,32)(28,29);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,27,26,32)(20,24,21,30)(23,28,31,29);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(19,26)(23,31)(27,32)(28,29);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19,21,29,22,26,20,28)(18,23,24,27,25,31,30,32);(2,6)(7,8)(17,20,22,21)(18,24,25,30)(19,28,26,29)(23,27,31,32)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fh,e^2=i,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=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308955, #72308965 |
| maximal normal subgroup of | #72308955, #72308965 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517410);