Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,9)(7,8,12,11)(13,14)(15,19)(16,17)(18,20)(21,22)(23,29)(24,33)(25,34)(26,27)(28,32)(30,31)(35,36);(6,9)(7,12)(13,15,18,17)(14,19,20,16)(22,27)(25,33)(28,29)(31,35);(5,7,10,12)(6,11,9,8)(13,16)(14,15)(17,20)(18,19)(21,23,26,32)(22,28,27,29)(24,30,34,36)(25,31,33,35);(1,2,3,4)(5,8)(6,7)(9,12)(10,11)(13,17,18,15)(14,19,20,16)(21,24)(22,25)(23,30)(26,34)(27,33)(28,31)(29,35)(32,36);(5,9,10,6)(7,8,12,11)(21,25)(22,24)(23,31)(26,33)(27,34)(28,30)(29,36)(32,35)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=f,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4894986);