Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1658 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)(7,11)(10,12)(14,18)(15,21)(16,23)(26,28)(29,30,34,35)(31,32,38,36)(33,41,42,37)(39,43,44,40);(5,6,9,8)(7,10,11,12)(14,18)(19,24)(20,25)(26,28)(30,35)(31,38)(33,42)(40,43);(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,38)(30,36,35,32)(33,39,42,44)(37,43,41,40);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,36)(30,31,35,38)(33,40,42,43)(37,39,41,44);(5,8,9,6)(7,12,11,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,37,35,41)(31,39,38,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=h,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4249335);