Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,21)(22,24)(25,26,28,30)(27,29,32,31);(9,11)(10,12)(13,15)(14,16)(18,21)(22,23)(25,26,28,30)(27,29,32,31);(2,5)(6,7)(10,14)(11,15)(17,18,20,21)(19,22,24,23)(25,26)(27,31)(28,30)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=f,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.
| maximal subgroup of | #72303671, #72303676, #72303692, … |
| maximal normal subgroup of | #72303671, #72303676, #72303692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213982);