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 | |
1796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(6,7)(8,12)(13,14,17,15)(16,19,20,18)(21,22)(23,25)(24,31)(26,27)(28,32)(29,34)(30,36)(33,35);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15,17,14)(16,19,20,18)(22,27)(23,29)(24,32)(33,36);(2,3)(5,6,9,7)(8,11,12,10)(21,23,26,29)(22,25,27,34)(24,30,32,35)(28,33,31,36);(5,7,9,6)(8,11,12,10)(13,16,17,20)(14,18,15,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,31)(29,35)(34,36);(1,3,4,2)(5,8,9,12)(6,10,7,11)(21,25)(22,23)(24,33)(26,34)(27,29)(28,30)(31,35)(32,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=fgi,e^2=g,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, 2284321);