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 | |
4417 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)(13,21)(17,22)(18,23)(20,24)(25,26)(27,35)(28,38)(29,34)(30,32)(31,37)(33,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,30,37)(26,31,32,35)(28,36,39,29)(33,40,38,34);(1,3,5,4)(2,7,6,8)(10,15)(12,19)(17,22)(20,24)(26,32)(31,35)(33,38)(34,40);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,28)(26,33)(27,36)(29,37)(30,39)(31,40)(32,38)(34,35);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,36)(26,34,32,40)(27,28,37,39)(31,33,35,38)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7624543);