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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,19)(12,21)(13,18)(14,22)(15,17)(16,20)(23,24)(25,26)(27,33)(28,30)(29,32)(31,35)(34,40)(36,38)(37,39);(1,2,5,4)(3,7,8,6)(9,11,15,20)(10,16,17,19)(12,13,22,23)(14,18,21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3,5,8)(2,6,4,7)(10,17)(11,20)(12,22)(18,24)(26,28)(31,33)(32,36)(39,40);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,13)(15,22)(16,18)(17,21)(19,24)(20,23)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,13)(10,18)(11,12)(14,16)(15,23)(17,24)(19,21)(20,22)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40);(9,14)(10,12)(11,18)(13,16)(15,21)(17,22)(19,23)(20,24)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9771107);