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 | |
2041 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)(5,6,9,7)(8,10,12,11)(13,14,17,15)(16,18,20,19)(22,25)(23,29)(24,32)(34,36);(6,7)(10,11)(14,15)(18,19)(21,22,26,25)(23,30,29,27)(24,28,32,33)(31,36,35,34);(8,12)(10,11)(13,14,17,15)(16,19,20,18)(21,23,26,29)(22,27,25,30)(24,31,32,35)(28,34,33,36);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,17,14)(16,18,20,19)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36);(5,8,9,12)(6,10,7,11)(13,16)(14,18)(15,19)(17,20)(21,25,26,22)(23,27,29,30)(24,28,32,33)(31,36,35,34)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=fg,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, 3261282);