Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,10,12,8)(13,14)(15,20)(16,18)(17,19)(21,22)(23,27)(24,25)(26,28);(5,7,9,12)(6,8,11,10)(13,15,18,17)(14,19,16,20)(22,25)(26,27);(14,16)(19,20)(21,22,24,25)(23,26,28,27);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,19,17,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,10)(6,7,11,12)(13,17,18,15)(14,19,16,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837802);