Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,23,24,22);(5,6)(7,8)(10,12)(14,15)(17,19,21,24)(18,22,20,23);(2,4)(6,8)(17,19)(18,23)(20,22)(21,24);(9,11)(10,14)(12,15)(13,16);(1,2)(3,4)(5,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^9|a^2=f,b^2=fi,e^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 | #72232038, #72232042, #72232062, … |
| maximal normal subgroup of | #72232038, #72232042, #72232062, … |
| maximal quotient of | #72323483, #72323556, C2×#4306532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213728);