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 | |
4331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,21)(14,16)(17,22)(19,23)(20,24)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,5,6)(3,7,8,4)(9,11,14,12)(10,15,16,18)(13,19,22,20)(17,23,21,24)(29,37)(32,36)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(10,16)(13,22)(15,18)(19,20)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,15)(12,18)(13,17)(14,16)(19,23)(20,24)(21,22)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,2)(3,4)(5,6)(7,8)(9,12,14,11)(10,15,16,18)(13,20,22,19)(17,23,21,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gh,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, 8838210);