Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,11)(10,12)(13,14)(15,18)(16,20)(17,19)(21,22,27,24)(23,29,25,28)(26,32,34,30)(31,35,33,36);(6,9)(10,12)(13,15,17,20)(14,18,19,16)(21,23,27,25)(22,28,24,29)(26,31,34,33)(30,35,32,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24,27,22)(23,29,25,28)(26,30,34,32)(31,35,33,36);(1,2,3,4)(21,25,27,23)(22,29,24,28)(26,33,34,31)(30,36,32,35);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,26,27,34)(22,30,24,32)(23,31,25,33)(28,35,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4323820);