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 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,35)(27,36,38,32)(28,30,34,39)(29,33,37,40);(2,3)(6,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,27)(26,32)(28,37)(29,34)(30,33)(31,38)(35,36)(39,40);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(16,21)(19,24);(1,4,5,8)(2,6,3,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,31,34)(26,30,35,39)(27,37,38,29)(32,33,36,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29,31,37)(26,33,35,40)(27,34,38,28)(30,36,39,32);(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30)(26,34)(27,33)(28,35)(29,32)(31,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9633644);