Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,12)(8,11)(9,10)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28);(6,9)(11,12)(14,16)(18,19)(21,22,24,26)(23,27,28,25);(1,3,4,2)(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,18,16,19)(21,22)(23,25)(24,26)(27,28);(5,8,10,7)(6,12,9,11);(5,9,10,6)(7,12,8,11)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fhi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 439728);