Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
2136 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)(9,10,14,15)(11,13,12,16)(17,18,22,24)(19,25,27,20)(21,23,30,32)(26,31,29,28);(6,8)(10,15)(13,16)(17,19,22,27)(18,20,24,25)(21,29,30,26)(23,31,32,28);(5,6,7,8)(9,11,14,12)(10,13,15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,32);(1,2)(3,4)(9,12,14,11)(10,16,15,13)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(9,13,14,16)(10,11,15,12)(17,21,22,30)(18,23,24,32)(19,26,27,29)(20,28,25,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1940802);