Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1686 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)(5,6)(7,11)(8,10)(9,12)(15,19)(18,20)(21,22,26,27)(23,24,29,32)(25,33,35,28)(30,34,36,31);(5,7,10,9)(6,11,8,12)(14,17)(18,20)(22,27)(23,29)(28,33)(30,36);(5,6,10,8)(7,12,9,11)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,30)(26,29)(27,32)(28,31)(33,34)(35,36);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(21,24)(22,23)(25,34)(26,32)(27,29)(28,36)(30,33)(31,35);(5,9,10,7)(6,11,8,12)(13,15,16,19)(14,18,17,20)(21,25,26,35)(22,28,27,33)(23,30,29,36)(24,31,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773441);