Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15172 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(11,15)(14,16)(17,18,23,28)(19,25,20,24)(21,29,31,26)(22,27,32,30);(10,13)(14,16)(17,19,23,20)(18,24,28,25)(21,27,31,30)(22,26,32,29);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,25,28,24)(21,27,31,30)(22,26,32,29);(1,2,3,4)(5,7,8,6)(17,21,23,31)(18,26,28,29)(19,27,20,30)(22,24,32,25);(9,11,12,15)(10,14,13,16)(17,22)(18,27)(19,29)(20,26)(21,24)(23,32)(25,31)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4250221);