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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1687 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)(5,6,9,11)(7,12,10,8)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,34,35)(31,37,39,33)(32,40,41,36)(38,43,42,44);(5,7,9,10)(6,8,11,12)(15,22)(18,21)(23,28)(25,27)(30,35)(32,41)(33,37)(38,42);(5,8,9,12)(6,10,11,7)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31)(30,33)(32,42)(34,39)(35,37)(36,44)(38,41)(40,43);(1,2,3,4)(5,8)(6,7)(9,12)(10,11)(13,15,17,22)(14,18,20,21)(16,23,24,28)(19,25,26,27)(29,32,34,41)(30,36,35,40)(31,38,39,42)(33,43,37,44);(5,7)(6,8)(9,10)(11,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33)(30,31)(32,43)(34,37)(35,39)(36,38)(40,42)(41,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=fg,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, 4407980);