Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,18,21)(12,19,20,15)(17,23,24,22);(1,3)(2,5)(4,7)(6,8)(14,21)(22,23);(5,8)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,23)(20,24);(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,22,21,23)Polycyclic presentation:
2^9|a^2=h,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72236668, #72236681, #72251254 |
| maximal normal subgroup of | #72236668, #72236681, #72251254 |
| maximal quotient of | #72257491, #72263810, #72263820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140740);