Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2796 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)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(18,19)(22,23)(25,26,29,30)(27,28,31,32);(10,13)(14,15)(17,18,21,19)(20,23,24,22)(25,27)(26,28)(29,31)(30,32);(2,5)(6,7)(10,13)(14,15)(26,30)(27,31);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^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 | #72260758, #72260843, #72260845, … |
| maximal normal subgroup of | #72260758, #72260843, #72260845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213985);