Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2180 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)(6,7)(8,12)(14,16)(15,19)(21,22,26,30)(23,27,28,24)(25,33,35,29)(31,32,36,34);(6,7)(10,11)(15,19)(18,20)(21,23,26,28)(22,24,30,27)(25,31,35,36)(29,32,33,34);(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,28)(29,34)(32,33)(35,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,24,26,27)(22,28,30,23)(25,34,35,32)(29,31,33,36);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,20,19,18)(21,25,26,35)(22,29,30,33)(23,31,28,36)(24,32,27,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2671572);