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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,15,19,16)(12,13,17,23)(20,21,24,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,36)(37,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,24)(22,23)(25,27)(26,28)(29,31)(30,36)(32,33)(34,37)(35,38)(39,40);(1,3,5,4)(2,6,7,8)(11,19)(15,16)(20,24)(21,22)(25,28,32,30)(26,33,36,27)(29,35,39,37)(31,38,40,34);(3,4)(6,8)(9,11)(10,16)(12,20)(13,22)(14,19)(15,18)(17,24)(21,23)(25,29,32,39)(26,34,36,38)(27,31,33,40)(28,37,30,35);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,20)(14,17)(15,21)(16,22)(18,23)(19,24)(25,30,32,28)(26,27,36,33)(29,35,39,37)(31,38,40,34);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,21)(12,18)(14,23)(15,24)(16,20)(19,22)(25,31)(26,35)(27,29)(28,38)(30,34)(32,40)(33,39)(36,37)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8640923);