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) = 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 | |
1892 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)(7,11)(8,9)(10,12)(21,22)(23,29)(24,33)(25,35)(26,27)(28,32)(30,31)(34,36);(7,12)(10,11)(13,14,17,16)(15,19,20,18)(22,27)(25,33)(28,29)(31,36);(6,9)(10,11)(13,15,17,20)(14,18,16,19)(21,23)(22,28)(24,34)(25,36)(26,32)(27,29)(30,35)(31,33);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,20,19)(21,24,26,35)(22,25,27,33)(23,30,32,34)(28,31,29,36);(5,7)(6,10)(8,12)(9,11)(13,16,17,14)(15,19,20,18)(21,25)(22,24)(23,31)(26,33)(27,35)(28,30)(29,34)(32,36)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=fh,e^2=h,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, 4067649);