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 | |
2040 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)(6,9)(7,11)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(24,31)(35,36);(6,9)(10,12)(15,19)(18,20)(21,22,25,26)(23,27,29,34)(24,28,31,32)(30,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,27)(24,32)(25,26)(28,31)(29,34)(30,36)(33,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,23,25,29)(22,27,26,34)(24,33,31,30)(28,36,32,35);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,30)(25,31)(26,32)(27,35)(29,33)(34,36)Polycyclic presentation:
2^9|a^2=gh,b^2=f,d^2=f,e^2=gh,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, 3257875);