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 | |
1772 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)(5,6,9,12)(7,10,8,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,32)(33,41)(34,35)(36,37)(38,42)(39,43)(40,44);(5,7,9,8)(6,10,12,11)(15,22)(18,21)(23,28)(26,27)(30,35)(31,37)(33,42)(40,43);(5,6)(7,10)(8,11)(9,12)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,41)(42,44);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,32,34,36)(30,37,35,31)(33,40,42,43)(38,44,41,39);(5,8,9,7)(6,11,12,10)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,38)(31,39)(32,40)(34,42)(35,41)(36,43)(37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fh,d^2=f,e^2=fg,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, 4220939);