Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,11,14,17,12,20,15,18,19,21,22,24,23,16);(1,3,5,4)(2,6,7,8)(10,11)(12,19)(13,23)(14,22)(16,17)(18,21)(20,24);(1,4,5,3)(2,6,7,8);(9,11,19,17,14,20,23,18,15,21,13,24,22,16,12,10)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=ghi,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72313208, #72313210, #72313218, … |
| maximal normal subgroup of | #72313208, #72313210, #72313218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405260);