Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1598 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)(13,14,17,20)(15,21,23,18)(16,24,25,19)(22,26,28,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(6,10)(11,12)(16,25)(19,24)(22,28)(26,27)(29,30,33,36)(31,34,39,37)(32,35,42,40)(38,43,41,44);(5,6,9,10)(7,11,8,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,30)(31,34)(32,40)(33,36)(35,42)(37,39)(38,44)(41,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(5,8,9,7)(6,12,10,11)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fgh,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4066287);