Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2056 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)(13,14,19,20)(15,16,17,18)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,6)(7,11)(8,10)(9,12)(13,15,19,17)(14,16,20,18)(22,26)(23,30)(28,32)(31,36);(5,6,8,10)(7,9,12,11)(13,16,19,18)(14,15,20,17)(21,22)(23,29)(24,32)(25,26)(27,30)(28,33)(31,34)(35,36);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,17,19,15)(14,16,20,18)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,35)(33,36);(5,7,8,12)(6,9,10,11)(13,18,19,16)(14,15,20,17)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=g,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5186374);