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) = 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 | |
2404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,29)(28,30)(31,32);(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,22,20)(18,23,24,21)(25,27,30,32)(26,31,28,29);(2,3)(6,7)(10,15)(11,12)(18,24)(19,20)(26,28)(29,31);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,16,15,14)(17,20,22,19)(18,23,24,21)(25,28,30,26)(27,29,32,31);(1,4,5,8)(2,6,3,7)(9,10)(11,16)(12,14)(13,15)(17,21,22,23)(18,19,24,20)(25,29)(26,32)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,d^2=g,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, 4070204);