Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,3,4)(5,6,9,7)(8,11,12,10)(15,24)(20,25)(26,37)(27,35)(28,38)(34,39)(40,43)(41,44);(6,7)(10,11)(13,14,19,23)(15,25,24,20)(16,29,32,21)(17,18,22,33)(26,34,37,39)(27,38,35,28)(30,42,36,31)(40,41,43,44);(5,7,9,6)(8,11,12,10)(13,15,19,24)(14,20,23,25)(16,26,32,37)(17,27,22,35)(18,28,33,38)(21,34,29,39)(30,40,36,43)(31,41,42,44);(5,8)(6,10)(7,11)(9,12)(13,16,19,32)(14,21,23,29)(15,26,24,37)(17,30,22,36)(18,31,33,42)(20,34,25,39)(27,40,35,43)(28,41,38,44);(13,17)(14,18)(15,27)(16,30)(19,22)(20,28)(21,31)(23,33)(24,35)(25,38)(26,40)(29,42)(32,36)(34,41)(37,43)(39,44);(13,18)(14,22)(15,28)(16,31)(17,23)(19,33)(20,35)(21,36)(24,38)(25,27)(26,41)(29,30)(32,42)(34,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534806);