Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,7)(8,12)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(5,6,9,7)(8,11,12,10)(13,15,16,19)(14,17,18,20)(23,29)(26,33)(30,35)(34,36);(5,7,9,6)(8,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,15)(14,17)(16,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4360338);