Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 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)(7,12)(8,10)(9,11)(13,14,19,24)(15,25,29,41)(16,22,31,21)(17,23,18,33)(20,27,36,28)(26,40,43,39)(30,38,32,37)(34,42,35,44);(7,8)(10,12)(13,15,17,27,19,29,18,28)(14,20,33,41,24,36,23,25)(16,26,32,44,31,43,30,42)(21,34,37,40,22,35,38,39);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,19,31)(14,21,24,22)(15,26,29,43)(17,32,18,30)(20,34,36,35)(23,38,33,37)(25,39,41,40)(27,44,28,42);(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,35)(21,24)(23,37)(25,40)(27,44)(28,42)(29,43)(33,38)(34,36)(39,41);(6,11)(10,12)(13,17,19,18)(14,23,24,33)(15,27,29,28)(16,30,31,32)(20,25,36,41)(21,37,22,38)(26,42,43,44)(34,40,35,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390573);