Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
2236 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)(7,8)(9,12)(10,11)(14,18)(15,19)(21,22,27,30)(23,28,24,33)(25,36,29,26)(31,32,34,35);(5,7)(6,10)(8,12)(9,11)(13,14)(15,16)(17,18)(19,20)(21,23,27,24)(22,28,30,33)(25,34,29,31)(26,32,36,35);(5,8,9,10)(6,11,12,7)(13,15,17,19)(14,16,18,20)(21,24,27,23)(22,28,30,33)(25,34,29,31)(26,35,36,32);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,16,17,20)(14,15,18,19)(21,25,27,29)(22,26,30,36)(23,31,24,34)(28,35,33,32);(5,8)(6,11)(7,12)(9,10)(13,15)(14,16)(17,19)(18,20)(21,26)(22,29)(23,32)(24,35)(25,30)(27,36)(28,31)(33,34)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3375444);