Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1598 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)(7,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,34,37)(31,38,35,32)(33,36,42,44)(39,43,41,40);(6,9)(10,12)(14,18)(16,24)(19,21)(22,28)(29,31,34,35)(30,32,37,38)(33,39,42,41)(36,40,44,43);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,25)(17,18)(20,24)(21,23)(22,27)(26,28)(29,31)(30,32)(33,41)(34,35)(36,43)(37,38)(39,42)(40,44);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32)(30,35)(31,37)(33,40)(34,38)(36,41)(39,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,41)(37,44)(38,43)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5388103);