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 | |
4386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,19)(15,20)(21,23)(22,24)(25,26,30,35)(27,31,37,39)(28,33,38,32)(29,34,36,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,22)(17,19)(18,24)(25,27,30,37)(26,31,35,39)(28,29,38,36)(32,40,33,34);(11,20)(15,17)(21,24)(22,23)(25,28,30,38)(26,32,35,33)(27,36,37,29)(31,34,39,40);(9,12,14,18)(10,16,19,13)(11,21,20,24)(15,22,17,23)(25,28)(26,33)(27,29)(30,38)(31,34)(32,35)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,19)(16,23)(18,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,13,14,16)(10,18,19,12)(11,22,20,23)(15,21,17,24)(25,29)(26,34)(27,28)(30,36)(31,33)(32,39)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7636293);