Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(10,13)(15,16)(17,18,21,22)(19,24,20,23)(25,26)(27,31)(28,32)(29,30);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,15,12,16)(18,22)(23,24)(25,26,29,30)(27,31,32,28);(9,11,14,12)(10,15,13,16)(17,19,21,20)(18,23,22,24)(25,27)(26,28)(29,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,28)(29,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,16,12,15)(17,20,21,19)(18,24,22,23)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fgi,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 811819);