Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24);(10,13)(14,15)(17,18,20,22)(19,21,24,23);(1,2,3,4)(5,7,8,6)(17,19)(18,21)(20,24)(22,23);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df176 conjugacy classes — view the full character table.
| maximal subgroup of | #72329463, #72329466, #72329467 |
| maximal normal subgroup of | #72329463, #72329466, #72329467 |
| maximal quotient of | #72331622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420946);