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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2043 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)(7,9)(8,10)(11,12)(14,18)(15,20)(21,22,25,28)(23,26,30,35)(24,31,33,27)(29,36,32,34);(6,7)(10,11)(13,14)(15,16)(17,18)(19,20)(21,23,25,30)(22,26,28,35)(24,32,33,29)(27,36,31,34);(8,12)(10,11)(14,18)(16,19)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,19,18,16)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,36)(32,33);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,15,18,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,35)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=i,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, 3380041);