Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26,28,30)(27,31,32,29);(2,6)(7,8)(9,11,14,12)(10,15,16,13)(17,19)(18,20)(21,24)(22,23);(1,3,5,4)(2,7,6,8)(9,11,14,12)(10,13,16,15)(17,20,23,21)(18,19,24,22);(1,4,5,3)(2,7,6,8)(9,12,14,11)(10,15,16,13)(17,21,23,20)(18,19,24,22)(25,27,28,32)(26,29,30,31);(1,2)(3,8)(4,7)(5,6)(9,13,14,15)(10,11,16,12)(17,22,23,19)(18,20,24,21)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=fi,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, 4214719);