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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2124 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)(5,6,8,9)(7,10,11,12)(13,14,18,20)(15,16,17,19)(22,24)(23,29)(25,33)(35,36);(6,9)(10,12)(13,15)(14,19)(16,20)(17,18)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,35,34,36);(7,11)(10,12)(13,16,18,19)(14,17,20,15)(21,23)(22,27)(24,30)(25,34)(26,29)(28,36)(31,33)(32,35);(1,2,3,4)(5,6)(7,12)(8,9)(10,11)(13,17,18,15)(14,16,20,19)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,35,34,36);(5,7,8,11)(6,10,9,12)(13,16)(14,17)(15,20)(18,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,35)(29,34)(30,36)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=fghi,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, 3386338);