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) = 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 | |
1891 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)(7,12)(8,11)(9,10)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30)(31,40)(32,36)(33,42)(34,41)(35,39)(37,43)(38,44);(6,10)(7,8)(15,21)(18,25)(22,27)(26,28)(29,31,35,32)(30,36,39,40)(33,38,43,41)(34,37,44,42);(5,7,9,8)(6,11,10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,35,31)(30,36,39,40)(33,41,43,38)(34,37,44,42);(1,2,3,4)(5,8,9,7)(6,11,10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,33,35,43)(30,37,39,42)(31,41,32,38)(34,40,44,36);(5,6)(7,12)(8,11)(9,10)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,34)(30,38)(31,37)(32,42)(33,40)(35,44)(36,43)(39,41)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=f,d^2=hi,e^2=gh,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, 5192129);