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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1742 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)(5,6,9,11)(7,10,12,8)(14,18)(15,21)(16,24)(26,28)(29,30,33,34)(31,37,38,35)(32,36,40,42)(39,44,41,43);(5,7,9,12)(6,8,11,10)(14,18)(16,24)(19,22)(23,27)(30,34)(31,38)(32,40)(43,44);(7,12)(8,10)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31,33,38)(30,35,34,37)(32,41,40,39)(36,44,42,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44);(5,8,9,10)(6,7,11,12)(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)(29,32)(30,36)(31,39)(33,40)(34,42)(35,43)(37,44)(38,41)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgh,d^2=gi,e^2=fg,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, 5188400);