Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(7,11)(10,12)(15,19)(17,20)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35);(6,9)(10,12)(22,26)(24,33)(27,29)(30,36);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,3,4,2)(13,15)(14,17)(16,19)(18,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(5,7,8,11)(6,10,9,12)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4238611);