Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2304 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)(5,6,9,12)(7,10,8,11)(15,19)(18,20)(21,22)(23,24)(25,33)(26,27)(28,32)(29,35)(30,34)(31,36);(5,7,9,8)(6,10,12,11)(14,17)(18,20)(22,27)(23,28)(29,33)(30,36);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,28)(22,24,27,32)(25,30,35,36)(29,31,33,34);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(21,24)(22,28)(23,27)(25,34)(26,32)(29,30)(31,35)(33,36);(5,8,9,7)(6,11,12,10)(13,15)(14,18)(16,19)(17,20)(21,25,26,35)(22,29,27,33)(23,30,28,36)(24,31,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4325532);