Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3280 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)(7,12)(8,10)(9,11)(13,14)(15,24)(16,22)(17,23)(18,32)(19,21)(20,26)(25,39)(27,35)(28,37)(29,36)(30,33)(31,38)(34,41)(40,43)(42,44);(1,2)(3,4)(13,15,18,26,19,27,17,30)(14,20,23,24,21,33,32,35)(16,25,28,41,29,42,31,40)(22,34,38,39,36,43,37,44);(2,4)(5,7,9,8)(6,10,11,12)(14,21)(20,33)(22,36)(23,32)(24,35)(34,43)(37,38)(39,44);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,29)(14,22,21,36)(15,25,27,42)(17,31,18,28)(20,34,33,43)(23,38,32,37)(24,39,35,44)(26,41,30,40);(13,17,19,18)(14,23,21,32)(15,26,27,30)(16,28,29,31)(20,35,33,24)(22,37,36,38)(25,40,42,41)(34,39,43,44)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359296);