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 | |
4434 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)(4,6)(5,8)(9,10,14,18)(11,12,16,15)(13,17,21,24)(19,20,23,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,14,16)(10,15,18,12)(13,19,21,23)(17,22,24,20)(26,27)(28,35)(32,34)(36,40);(1,2)(3,5)(4,6)(7,8)(13,21)(17,24)(19,23)(20,22)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(9,12)(10,16)(11,18)(13,20)(14,15)(17,23)(19,24)(21,22)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(9,13,14,21)(10,17,18,24)(11,19,16,23)(12,20,15,22)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8325614);