Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(13,14)(15,19)(16,20)(17,18)(21,22,26,27)(23,25,29,34)(24,31,32,28)(30,36,35,33);(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,20,16)(22,27)(23,29)(24,32)(33,36);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,16,18,19)(21,23)(22,25)(24,30)(26,29)(27,34)(28,33)(31,36)(32,35);(1,2,3,4)(21,24,26,32)(22,28,27,31)(23,30,29,35)(25,33,34,36);(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,19)(18,20)(21,25)(22,29)(23,27)(24,33)(26,34)(28,35)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2223180);