Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1741 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)(5,6,9,7)(8,10,12,11)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(33,41)(36,40)(38,44)(42,43);(14,19)(15,22)(17,25)(24,27)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(6,7)(10,11)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,8,9,12)(6,10,7,11)(13,17,18,25)(14,21,19,26)(15,23,22,28)(16,24,20,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843811);