Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | #33717 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
743 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)(3,6)(4,7)(5,8)(10,15)(11,18,21,17)(12,22)(13,16)(19,23,24,20);(1,3,6,2)(4,7,5,8)(9,10,16,22,14,12,13,15)(11,19,20,18,21,24,23,17);(1,4,6,5)(2,7,3,8)(9,11,10,17,13,20,15,19,14,21,12,18,16,23,22,24);(1,5,6,4)(2,7,3,8)(9,12,13,22,14,10,16,15)(11,18,20,24,21,17,23,19)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=efh,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264735, #72264781, #72265009, … |
| maximal normal subgroup of | #72264735, #72264781, #72265009, … |
| maximal quotient of | #72309582, #72309771, #72310261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404342);