Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,10)(7,8,12,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28)(29,30,34,37)(31,32,39,35)(33,36,42,44)(38,40,43,41);(6,10)(7,12)(13,14,17,20)(15,18,23,21)(16,19,25,24)(22,26,28,27)(29,31,34,39)(30,35,37,32)(33,38,42,43)(36,41,44,40);(6,10)(8,11)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28)(29,32,34,35)(30,31,37,39)(33,41,42,40)(36,43,44,38);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,41)(40,44)(42,43);(5,8,9,11)(6,7,10,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)(29,33,34,42)(30,36,37,44)(31,38,39,43)(32,40,35,41)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507001);