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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2127 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)(5,6,8,10)(7,11,12,9)(13,14,18,17)(15,19,20,16)(21,22,26,29)(23,24,31,27)(25,28,33,35)(30,32,36,34);(5,7,8,12)(6,9,10,11)(13,15,18,20)(14,16,17,19)(25,33)(28,35)(30,36)(32,34);(6,10)(9,11)(14,17)(15,20)(21,23,26,31)(22,27,29,24)(25,30,33,36)(28,34,35,32);(1,2,3,4)(5,7)(6,11)(8,12)(9,10)(13,16)(14,15)(17,20)(18,19)(21,24,26,27)(22,23,29,31)(25,32,33,34)(28,30,35,36);(5,6)(7,11)(8,10)(9,12)(13,17,18,14)(15,16,20,19)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=i,e^2=h,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, 5188609);