Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1850 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)(7,11)(8,12)(9,10)(13,14,18,20)(15,17,19,16)(21,22,26,27)(23,28,31,35)(24,25,30,29)(32,34,36,33);(5,7)(6,8)(9,12)(10,11)(13,15,18,19)(14,16,20,17)(22,27)(23,31)(24,30)(33,34);(6,10)(8,11)(13,16,18,17)(14,19,20,15)(21,23)(22,28)(24,32)(25,34)(26,31)(27,35)(29,33)(30,36);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,17,18,16)(14,19,20,15)(21,24,26,30)(22,29,27,25)(23,32,31,36)(28,33,35,34);(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,29)(22,30,27,24)(23,33,31,34)(28,32,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2216751);