Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1737 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)(6,10)(7,12)(13,14,18,19)(15,22,20,16)(17,25,27,21)(23,24,28,26)(29,30)(31,35)(32,39)(33,41)(34,37)(36,42)(38,40)(43,44);(14,19)(15,20)(21,25)(23,28)(29,31,34,39)(30,35,37,32)(33,38,42,44)(36,43,41,40);(5,6,9,10)(7,11,12,8)(13,15,18,20)(14,16,19,22)(17,23,27,28)(21,24,25,26)(29,31)(30,32)(33,38)(34,39)(35,37)(36,40)(41,43)(42,44);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,20)(15,19)(17,26)(18,22)(21,23)(24,27)(25,28)(29,32)(30,31)(33,40)(34,35)(36,38)(37,39)(41,44)(42,43);(5,8,9,11)(6,7,10,12)(13,17,18,27)(14,21,19,25)(15,23,20,28)(16,24,22,26)(29,33,34,42)(30,36,37,41)(31,38,39,44)(32,40,35,43)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4342045);