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 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,9)(7,11)(13,14)(15,16)(17,18)(19,20)(21,22,26,24)(23,28,31,27)(25,29,34,32)(30,36,33,35);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,17)(18,20)(21,23,26,31)(22,27,24,28)(25,30,34,33)(29,35,32,36);(5,6)(7,12)(8,9)(10,11)(13,16,18,19)(14,20,17,15)(21,24,26,22)(23,27,31,28)(25,29,34,32)(30,36,33,35);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,28)(24,27)(25,33)(26,31)(29,35)(30,34)(32,36);(5,7)(6,10)(8,11)(9,12)(13,17,18,14)(15,16,20,19)(21,25)(22,29)(23,30)(24,32)(26,34)(27,35)(28,36)(31,33)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=fhi,e^2=fgh,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, 3386084);