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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2095 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)(13,14,17,18)(15,20,19,16)(21,22)(23,28)(24,31)(25,34)(26,27)(29,33)(30,36)(32,35);(8,12)(10,11)(14,18)(15,19)(21,23,27,25)(22,28,26,34)(24,32,33,30)(29,36,31,35);(6,7)(10,11)(15,19)(16,20)(21,24,27,33)(22,29,26,31)(23,30,25,32)(28,35,34,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,16,18,20)(21,25,27,23)(22,28,26,34)(24,30,33,32)(29,36,31,35);(5,8,9,12)(6,10,7,11)(13,16)(14,19)(15,18)(17,20)(21,26,27,22)(23,28,25,34)(24,31,33,29)(30,35,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=i,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, 5187932);