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 | |
4646 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)(11,16)(12,13)(14,15)(17,22)(18,19)(20,21)(23,24)(25,26)(27,31)(28,33)(29,34)(30,35)(32,37)(36,39)(38,40);(1,3,5,4)(2,6,7,8)(10,15)(12,20)(16,22)(18,23)(25,27)(26,31)(28,34)(29,33)(30,37)(32,35)(36,38)(39,40);(1,3,5,4)(2,6,7,8)(11,17)(16,22)(18,23)(19,24)(25,27,30,37)(26,31,35,32)(28,36,38,34)(29,33,39,40);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,23)(21,24)(25,27)(26,32)(28,36)(29,33)(30,37)(31,35)(34,38)(39,40);(1,4,5,3)(2,6,7,8)(9,12,14,20)(10,13,15,21)(11,18,17,23)(16,19,22,24)(25,28)(26,33)(27,36)(29,32)(30,38)(31,39)(34,37)(35,40);(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,19)(14,21)(15,20)(16,18)(17,24)(22,23)(25,29)(26,34)(27,33)(28,31)(30,39)(32,38)(35,36)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877288);