Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2404 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(14,22)(17,24);(1,2,4,5)(3,6,7,8)(9,10)(11,17,18,24)(12,16)(13,15)(14,19,22,20)(21,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,20)(13,18)(15,22)(16,24)(17,23)(19,21);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=e,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,g^c=gh,f^d=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279774, #72280527, #72280571, … |
| maximal normal subgroup of | #72279774, #72280527, #72280571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 158989);