Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2196 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)(5,6,10,8)(7,12,9,11)(13,14,18,19)(15,20,17,16)(21,22,26,27)(23,24,30,28)(25,33,34,29)(31,36,35,32);(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,16,19,20)(22,27)(23,30)(25,34)(32,36);(7,9)(11,12)(13,16,18,20)(14,15,19,17)(21,23,26,30)(22,28,27,24)(25,35,34,31)(29,32,33,36);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(13,14)(15,16)(17,20)(18,19)(21,24,26,28)(22,23,27,30)(25,36,34,32)(29,35,33,31);(5,9,10,7)(6,11,8,12)(13,17,18,15)(14,16,19,20)(21,25,26,34)(22,29,27,33)(23,31,30,35)(24,32,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4049424);