Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(15,18)(16,20)(21,22,25,26)(23,27,29,34)(24,31,33,28)(30,36,32,35);(5,7,10,9)(6,11,8,12)(13,14,17,19)(15,16,18,20)(22,26)(23,29)(28,31)(30,32);(1,2,3,4)(5,6,10,8)(7,11,9,12)(13,15,17,18)(14,16,19,20)(21,23)(22,27)(24,32)(25,29)(26,34)(28,36)(30,33)(31,35);(5,8,10,6)(7,11,9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36);(5,9,10,7)(6,11,8,12)(21,24)(22,28)(23,30)(25,33)(26,31)(27,35)(29,32)(34,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4293594);