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 | |
2052 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)(5,6,10,11)(7,9,8,12)(15,19)(18,20)(21,22)(23,25)(24,31)(26,27)(28,32)(29,35)(30,36)(33,34);(5,7,10,8)(6,9,11,12)(14,17)(18,20)(22,27)(23,29)(24,32)(34,36);(6,11)(7,8)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,25,27,35)(24,33,32,30)(28,36,31,34);(1,2,3,4)(5,8,10,7)(6,9,11,12)(21,24)(22,28)(23,30)(25,34)(26,32)(27,31)(29,33)(35,36);(5,9)(6,7)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,25)(22,23)(24,34)(26,35)(27,29)(28,30)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=fi,e^2=fgh,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, 5189079);