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) = C26×C4 |
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 | |
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)(6,9)(10,11)(14,18)(15,21)(20,25)(22,27)(29,30)(31,35)(32,40)(33,34)(36,42)(37,39)(38,44)(41,43);(5,6,8,9)(7,11,12,10)(15,21)(19,24)(22,27)(26,28)(30,33)(31,37)(36,40)(38,41);(5,6,8,9)(7,10,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,37)(30,35,33,39)(32,41,42,38)(36,44,40,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,41)(39,44);(5,7)(6,10)(8,12)(9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,30)(31,39,37,35)(32,40,42,36)(38,44,41,43)Polycyclic presentation:
2^9|b^2=fh,c^2=gh,d^2=fi,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, 4408022);