Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2400 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)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,24,20)(25,26)(27,29)(28,32)(30,31);(2,3)(6,7)(9,11,12,16)(10,13,14,15)(17,19,21,24)(18,20,22,23)(26,31)(27,28);(4,8)(6,7)(10,14)(13,15)(18,22)(19,24)(25,27,30,28)(26,29,31,32);(1,3,5,2)(4,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28,30,27)(26,32,31,29);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(25,29,30,32)(26,28,31,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=ghi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038078);