Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,17)(12,20,21,15)(13,16)(14,19,22,24)(18,23);(5,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24);(2,4)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=ci,f^b=fh,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235244, #72235250, #72235479, … |
| maximal normal subgroup of | #72235244, #72235250, #72235479, … |
| maximal quotient of | #72235854, #72250038, #72258393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53655);