Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,10,8,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,35,38)(31,33,32,36)(34,42,43,37)(39,44,40,41);(2,4)(14,18)(16,24)(19,21)(22,28)(29,31,35,32)(30,36,38,33)(34,40,43,39)(37,41,42,44);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32,35,31)(30,33,38,36)(34,40,43,39)(37,41,42,44);(5,6)(7,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,33)(30,32)(31,38)(34,41)(35,36)(37,40)(39,42)(43,44);(5,8,9,7)(6,11,12,10)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,34)(30,37)(31,39)(32,40)(33,41)(35,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=ghi,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1754417);