Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,14,17,12,16,13,22)(11,18,21,24,20,15,19,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15,16,18)(12,20)(13,19)(14,21)(17,23,22,24);(2,6)(5,8)(11,19)(13,14)(15,23)(17,22)(18,24)(20,21);(3,7)(5,8)(10,16)(11,20)(13,14)(15,18)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338922, #72338926, #72338927 |
| maximal normal subgroup of | #72338922, #72338926, #72338927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386934);