Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 2097152 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,16,19,18)(21,22,26,27)(23,25,30,28)(24,29,33,35)(31,34,36,32);(1,2,3,4)(5,6,8,10)(7,9,12,11)(15,19)(16,18)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,19)(14,18,20,16)(21,23,26,30)(22,28,27,25)(24,31,33,36)(29,32,35,34);(5,6)(7,9)(8,10)(11,12)(13,16,17,18)(14,19,20,15)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,32)(30,36);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,25,26,28)(22,23,27,30)(24,32,33,34)(29,36,35,31)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3471377);