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) = C29 |
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 | |
5274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,23,24,22);(2,3)(5,6)(7,8)(17,19,21,24)(18,22,20,23);(2,3)(6,7)(10,13)(14,15)(17,19)(18,23)(20,22)(21,24);(9,11)(10,14)(12,16)(13,15);(1,3,4,2)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,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 | #72261109, #72261122, #72261232, … |
| maximal normal subgroup of | #72261109, #72261122, #72261232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213770);