Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,26)(25,38)(27,34)(28,36)(29,37)(30,42)(31,35)(33,40)(39,44)(41,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,27,17,31)(14,20,22,24,23,35,32,34)(16,25,30,40,28,39,29,41)(21,33,37,38,36,43,42,44);(1,2,3,4)(6,10)(7,11)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(7,11)(9,12)(16,28)(21,36)(25,39)(29,30)(33,43)(37,42)(38,44)(40,41);(13,17,19,18)(14,22,23,32)(15,26,27,31)(16,29,28,30)(20,34,35,24)(21,37,36,42)(25,40,39,41)(33,44,43,38)Polycyclic presentation:
2^9|b^2=f,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366850);