Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| 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 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(14,18)(16,24)(19,21)(22,28)(29,30,33,34)(31,35,39,37)(32,36,41,40)(38,42,44,43);(5,6,9,8)(7,10,11,12)(14,18)(19,21)(20,25)(26,27)(30,34)(35,37)(36,40)(42,43);(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30)(31,37)(32,40)(33,34)(35,39)(36,41)(38,42)(43,44);(1,2)(3,4)(5,8,9,6)(7,10,11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928643);