Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7513980 |
| 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 | #23965 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1601 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)(10,14)(11,12)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19)(18,22)(20,25)(21,27)(23,29)(24,31)(26,28)(30,32);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16);(9,11,13,12)(10,16,14,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(2,7)(6,8)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7493948);