Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1666 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)(8,9)(10,12)(15,19)(17,20)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(5,7,9,12)(6,10,8,11)(13,14,16,18)(15,17,19,20)(24,32)(27,34)(30,36)(33,35);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7)(6,11)(8,10)(9,12)(13,15,16,19)(14,17,18,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fg,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5531552);