Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1780 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)(7,8)(11,12)(15,16)(19,20)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31);(5,6,10,9)(7,11,8,12)(14,18)(19,20)(22,27)(23,28)(25,34)(31,36);(5,7,10,8)(6,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15,17,16)(14,19,18,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36);(5,9,10,6)(7,12,8,11)(13,16,17,15)(14,20,18,19)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4226151);