Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1816 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)(5,6,8,11)(7,10,12,9)(13,14,18,19)(15,20,17,16)(21,22)(23,29)(24,28)(25,26)(27,30)(31,36)(32,34)(33,35);(5,7,8,12)(6,9,11,10)(13,15)(14,16)(17,18)(19,20)(22,26)(23,30)(24,32)(35,36);(5,7,8,12)(6,10,11,9)(13,16,18,20)(14,15,19,17)(21,23)(22,27)(24,33)(25,30)(26,29)(28,36)(31,32)(34,35);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,17,18,15)(14,16,19,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36);(5,6)(7,9)(8,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24,25,32)(22,28,26,34)(23,31,30,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5124258);