Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #264043 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,10)(7,8)(11,12)(13,14)(15,23)(16,26)(17,28)(18,20)(19,25)(21,27)(22,24);(6,11)(7,8)(10,12)(13,15,17,25,18,16,24,27)(14,19,22,23,20,21,28,26);(1,2,3,4)(5,6,7,10,9,12,8,11)(14,20)(19,21)(22,28)(23,26);(13,16,17,27,18,15,24,25)(14,21,22,26,20,19,28,23)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207731);