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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,10,8,11)(14,18)(16,23)(19,24)(22,27)(29,30,34,35)(31,32,39,36)(33,41,42,37)(38,43,44,40);(1,2,3,4)(15,21)(19,24)(22,27)(26,28)(30,35)(32,36)(33,42)(38,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,39)(30,36,35,32)(33,38,42,44)(37,43,41,40);(5,6)(7,10)(8,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,36)(30,31,35,39)(33,40,42,43)(37,38,41,44);(5,8,9,7)(6,11,12,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,37)(31,38)(32,40)(34,42)(35,41)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fgi,d^2=g,e^2=fi,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, 1696290);