Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2076 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)(7,11)(8,10)(9,12)(15,16)(18,19)(21,22,26,27)(23,24,29,28)(25,33,36,30)(31,35,34,32);(5,7,10,9)(6,11,8,12)(13,14,17,20)(15,18,16,19)(22,27)(23,29)(30,33)(31,34);(5,6,10,8)(7,12,9,11)(13,15,17,16)(14,18,20,19)(21,23,26,29)(22,28,27,24)(25,34,36,31)(30,32,33,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,28)(22,29,27,23)(25,35,36,32)(30,31,33,34);(5,9,10,7)(6,11,8,12)(13,16,17,15)(14,19,20,18)(21,25,26,36)(22,30,27,33)(23,31,29,34)(24,32,28,35)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3248812);