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 | |
2104 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)(5,6,8,9)(7,10,11,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,29)(25,30)(27,32)(28,34)(31,35)(33,36);(6,9)(10,12)(13,15,18,19)(14,17,20,16)(22,23)(27,29)(28,30)(35,36);(7,11)(10,12)(13,16,18,17)(14,15,20,19)(21,23,26,22)(24,29,32,27)(25,28,34,30)(31,35,33,36);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,17,18,16)(14,19,20,15)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=h,d^2=h,e^2=g,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, 3255850);