Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14722 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2194 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)(9,10)(11,16)(12,15)(13,14)(18,22)(20,24)(27,31)(30,32);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,15,12,16)(18,22)(19,23)(26,29)(30,32);(4,8)(6,7)(9,11,14,12)(10,15,13,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,13,16)(17,19)(18,20)(21,23)(22,24);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,15,12,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4538490);