Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,21)(16,18)(17,20)(19,23)(22,27)(24,25)(26,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,6)(7,9)(8,11)(10,12)(14,16)(17,25)(19,21)(22,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(6,7)(10,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8,9,12)(6,10,7,11)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=gh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4381342);