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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1669 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)(7,8,12,10)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24)(33,41)(36,40)(38,44)(42,43);(5,7)(6,10)(8,11)(9,12)(15,20)(16,19)(23,28)(24,27)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,9,12)(6,10,11,8)(13,15,18,20)(14,19,22,16)(17,23,26,28)(21,27,25,24)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7)(6,8)(9,12)(10,11)(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=fh,e^2=fg,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, 4509268);