Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1770 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)(7,11)(10,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(29,30,33,34)(31,35,39,37)(32,40,42,36)(38,44,41,43);(6,9)(10,12)(14,15)(16,23)(17,25)(26,27)(30,34)(35,37)(36,40)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31)(30,35)(32,41)(33,39)(34,37)(36,44)(38,42)(40,43);(1,2,3,4)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,44)(40,42)(41,43);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32)(30,36)(31,38)(33,42)(34,40)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fg,c^2=g,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188612);