Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,8)(7,11,9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,37)(32,34)(33,40)(35,39)(36,41)(38,42)(43,44);(5,7,10,9)(6,11,8,12)(15,22)(18,21)(23,28)(26,27)(30,32)(33,41)(35,37)(38,44);(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fh,d^2=fgi,e^2=fg,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, 1652915);