Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(9,10,12,13)(11,15,16,14)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(21,29)(24,28)(26,32)(30,31);(5,6)(7,8)(10,13)(11,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(9,11)(10,14)(12,16)(13,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4040951);