Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1614 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)(14,18)(15,21)(16,23)(26,28)(29,30,34,37)(31,38,36,33)(32,40,42,35)(39,43,44,41);(6,9)(10,12)(15,21)(19,24)(22,27)(26,28)(29,31,34,36)(30,33,37,38)(32,39,42,44)(35,43,40,41);(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,33)(32,39)(34,36)(35,43)(37,38)(40,41)(42,44);(1,2,3,4)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,42)(30,35,37,40)(31,39,36,44)(33,43,38,41);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,37)(32,41)(34,38)(35,39)(40,44)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3381500);