Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1687 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)(7,12)(8,9)(10,11)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(5,7,10,9)(6,8,11,12)(14,15)(19,21)(20,22)(26,27)(31,37)(35,38)(39,43)(42,44);(6,11)(7,9)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(5,9,10,7)(6,8,11,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fg,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513488);