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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1728 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)(5,6,10,12)(7,11,9,8)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(5,7,10,9)(6,11,12,8)(14,18)(16,24)(19,21)(22,28)(31,38)(34,37)(39,44)(42,43);(5,8,10,11)(6,9,12,7)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,31)(30,34)(32,39)(33,38)(35,42)(36,37)(40,43)(41,44);(1,2,3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,30)(31,37)(32,35)(33,36)(34,38)(39,43)(40,41)(42,44);(5,9,10,7)(6,8,12,11)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,32,33,41)(30,35,36,40)(31,39,38,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=hi,e^2=g,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, 4894428);