Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| 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×#120 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14,13,15,16,11)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(1,3,4,7)(2,5,6,8)(10,11)(12,16)(14,15)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,28)(29,32);(2,6)(5,8)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(9,11,12,10,13,14,16,15)(20,28)(23,32)(25,27)(29,31);(18,24)(19,26)(23,32)(25,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6947987);