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 | #16508 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2370 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)(7,11)(8,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,26,27)(23,28,29,24)(25,33,35,30)(31,32,34,36);(5,7,10,8)(6,11,12,9)(13,15,17,20)(14,18,16,19)(22,27)(24,28)(30,33)(32,36);(6,12)(7,8)(14,16)(18,19)(21,23)(22,28)(24,27)(25,34)(26,29)(30,32)(31,35)(33,36);(1,2,3,4)(5,8,10,7)(6,11,12,9)(13,16,17,14)(15,19,20,18)(21,24,26,28)(22,29,27,23)(25,32,35,36)(30,34,33,31);(5,9,10,11)(6,7,12,8)(13,15)(14,18)(16,19)(17,20)(21,25,26,35)(22,30,27,33)(23,31,29,34)(24,32,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=fh,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, 5188524);