Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,15,20)(11,21,23,24)(12,22,16,17)(13,19,14,18);(9,11,15,23)(10,16,20,12)(13,22,14,17)(18,21,19,24);(1,2)(3,6)(4,8)(5,7);(9,12)(10,17)(11,18)(13,24)(14,21)(15,16)(19,23)(20,22);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,20,19)(11,22,23,17)(12,24,16,21)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72338403, #72338417, #72338427, … |
| maximal normal subgroup of | #72338403, #72338417, #72338427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269835);