Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,26,28,24)(21,30,23,25)(27,29,32,31);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,26)(21,29)(22,28)(23,31)(25,32)(27,30);(2,6)(5,7)(10,13)(14,16)(17,20,22,18)(19,27,28,32)(21,30,23,25)(24,31,26,29);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,20)(24,26)(25,30)(27,32);(3,8)(5,7)(17,21)(18,25)(19,28)(20,30)(22,23)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308633, #72308985 |
| maximal normal subgroup of | #72308633, #72308985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6607982);