Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 6291456 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1872 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,14,16,12)(17,18,22,23)(19,24,20,21)(27,31)(29,32);(9,11)(10,14)(12,15)(13,16)(17,19,22,20)(18,21,23,24)(25,26,28,30)(27,29,31,32);(2,6)(4,7)(10,15)(11,16)(17,19,22,20)(18,21,23,24);(1,3,5,8)(2,7,6,4)(9,12,13,14)(10,11,15,16)(17,20,22,19)(18,21,23,24)(25,27)(26,29)(28,31)(30,32);(1,4,5,7)(2,8,6,3)(9,11)(10,14)(12,15)(13,16)(17,21)(18,19)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214637);