Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,31)(28,29)(30,34)(32,37)(33,40)(35,36)(38,39);(2,6)(3,7)(10,12)(15,19)(16,20)(22,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,34)(33,38)(39,40);(1,3)(2,4)(5,7)(6,8)(11,17)(13,21)(15,19)(16,20)(25,27,30,37)(26,32,34,31)(28,35,33,38)(29,39,40,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28,30,33)(26,29,34,40)(27,35,37,38)(31,36,32,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,36)(28,34)(30,40)(31,38)(32,35)(37,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838002);