Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
2126 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)(7,8)(9,11)(10,12)(14,19)(16,20)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35);(5,7,9,12)(6,10,11,8)(13,14,18,19)(15,20,17,16)(23,30)(26,29)(31,36)(34,35);(5,6)(7,8)(9,11)(10,12)(13,15,18,17)(14,20,19,16)(21,22)(23,26)(24,27)(25,28)(29,30)(31,34)(32,33)(35,36);(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,15)(17,19)(18,20)(21,23,25,30)(22,26,28,29)(24,31,33,36)(27,34,32,35);(5,8,9,10)(6,7,11,12)(13,17,18,15)(14,20,19,16)(21,24,25,33)(22,27,28,32)(23,31,30,36)(26,34,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2833405);