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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,11)(7,9,8,12)(13,14)(15,19)(16,20)(17,18)(22,26)(23,29)(24,32)(35,36);(6,11)(9,12)(14,18)(15,20)(21,22,25,26)(23,30,29,27)(24,33,32,28)(31,35,34,36);(5,7,10,8)(6,9,11,12)(13,15,17,20)(14,19,18,16)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,33)(34,36);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,16)(14,15)(17,19)(18,20)(21,23,25,29)(22,27,26,30)(24,34,32,31)(28,36,33,35);(5,9)(6,8)(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,24,25,32)(22,28,26,33)(23,31,29,34)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fg,d^2=ghi,e^2=g,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, 2833167);