Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(15,19)(17,20)(21,22,26,29)(23,30,33,27)(24,25,35,28)(31,36,34,32);(5,7)(6,9)(8,12)(10,11)(21,23,26,33)(22,27,29,30)(24,31,35,34)(25,32,28,36);(2,3)(6,8)(7,11)(13,14,16,18)(15,17,19,20)(21,23)(22,27)(24,34)(25,36)(26,33)(28,32)(29,30)(31,35);(1,3,4,2)(5,8,10,6)(7,12,11,9)(13,15)(14,17)(16,19)(18,20)(21,24,26,35)(22,28,29,25)(23,31,33,34)(27,36,30,32);(5,9,10,12)(6,11,8,7)(21,25,26,28)(22,24,29,35)(23,32,33,36)(27,31,30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942228);