Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,17)(15,16,20,19)(21,22,26,27)(23,29,31,24)(25,33,34,28)(30,32,36,35);(2,4)(5,7,9,12)(6,10,11,8)(13,15,18,20)(14,19,17,16)(22,27)(24,29)(25,34)(30,36);(1,2,3,4)(13,16)(14,20)(15,17)(18,19)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(5,6)(7,10)(8,12)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,35)(26,29)(27,31)(28,36)(30,33)(32,34);(5,8,9,10)(6,7,11,12)(13,17,18,14)(15,19,20,16)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2800027);