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 | |
2089 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)(7,10)(8,12)(9,11)(13,14)(15,16)(17,18)(19,20)(22,26)(23,29)(28,33)(31,35);(5,7,9,12)(6,8,11,10)(13,14,17,18)(15,19,20,16)(24,32)(28,33)(31,35)(34,36);(5,7,9,12)(6,10,11,8)(13,15,17,20)(14,16,18,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8,9,10)(6,7,11,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=gi,e^2=fh,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, 5188721);