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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2174 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)(7,10)(8,12)(9,11)(14,18)(16,20)(21,22,25,26)(23,27,30,34)(24,31,33,28)(29,36,32,35);(1,2,3,4)(5,7,9,12)(6,10,11,8)(14,18)(15,19)(22,26)(27,34)(28,31)(35,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14,17,18)(15,16,19,20)(21,22)(23,27)(24,31)(25,26)(28,33)(29,36)(30,34)(32,35);(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,16,18,20)(21,23,25,30)(22,27,26,34)(24,32,33,29)(28,36,31,35);(5,8)(6,7)(9,10)(11,12)(13,16)(14,19)(15,18)(17,20)(21,24,25,33)(22,28,26,31)(23,29,30,32)(27,35,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=fgi,d^2=gi,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, 3471358);