Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,9,12)(7,10,11,8)(15,21)(16,23)(19,24)(20,25)(29,30)(31,35)(32,40)(33,34)(36,42)(37,39)(38,44)(41,43);(5,7,9,11)(6,10,12,8)(14,18)(19,24)(20,25)(26,28)(30,33)(31,37)(36,40)(38,41);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,37)(30,35,33,39)(32,41,42,38)(36,44,40,43);(5,8,9,10)(6,11,12,7)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,41)(39,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,30)(31,39,37,35)(32,40,42,36)(38,44,41,43)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4255838);