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 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,8)(11,12)(13,14,18,15)(16,19,24,21)(17,20,25,22)(23,26,28,27)(30,34)(35,37)(36,40)(42,43);(6,10)(11,12)(14,15)(19,21)(20,22)(26,27)(29,30,33,34)(31,37,39,35)(32,36,41,40)(38,43,44,42);(5,6)(7,11)(8,12)(9,10)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,39)(38,43)(42,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(5,8,9,7)(6,12,10,11)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=ghi,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, 4521936);