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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1996 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)(7,12)(8,10)(13,14,19,20)(15,16,17,18)(21,22)(23,24)(25,33)(26,27)(28,31)(29,35)(30,36)(32,34);(5,6,9,11)(7,8,12,10)(13,15,19,17)(14,16,20,18)(22,27)(24,31)(29,33)(32,36);(5,7,9,12)(6,10,11,8)(13,16,19,18)(14,15,20,17)(21,23)(22,24)(25,34)(26,28)(27,31)(29,36)(30,35)(32,33);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,17,19,15)(14,16,20,18)(21,24)(22,28)(23,27)(25,32)(26,31)(29,34)(30,33)(35,36);(5,8)(6,7)(9,10)(11,12)(13,18,19,16)(14,15,20,17)(21,25,26,35)(22,29,27,33)(23,30,28,34)(24,32,31,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=fhi,e^2=fgh,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, 5192275);