Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(15,19)(18,20)(21,22,27,29)(23,30,24,25)(26,28,35,36)(31,34,32,33);(2,4)(7,11)(10,12)(21,23,27,24)(22,25,29,30)(26,31,35,32)(28,33,36,34);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,24,27,23)(22,25,29,30)(26,31,35,32)(28,34,36,33);(5,7,8,11)(6,10,9,12)(13,15,17,19)(14,18,16,20)(21,25)(22,23)(24,29)(26,34)(27,30)(28,32)(31,36)(33,35);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,20,19,18)(21,26)(22,28)(23,31)(24,32)(25,33)(27,35)(29,36)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=i,e^2=gi,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, 3700336);