Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(7,11)(9,12)(13,14,18,20)(15,19,16,17)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,6,8,10)(7,9,11,12)(13,15,18,16)(14,19,20,17)(22,26)(24,32)(27,29)(30,36);(5,6)(7,9)(8,10)(11,12)(13,16,18,15)(14,19,20,17)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,2)(3,4)(13,17,18,19)(14,16,20,15)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,7)(6,9)(8,11)(10,12)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848585);