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 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,20)(13,17)(14,18)(15,21)(16,22)(23,24)(25,26,30,34)(27,33,38,31)(28,32,39,29)(35,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,19)(12,17,22,24)(13,20,23,16)(25,27)(26,31)(28,37)(29,36)(30,38)(32,40)(33,34)(35,39);(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,23,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,36)(33,40)(37,38);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,29,30,32)(26,28,34,39)(27,36,38,40)(31,37,33,35);(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,23)(21,24)(25,28)(26,32)(27,37)(29,34)(30,39)(31,40)(33,36)(35,38);(9,13)(10,17)(11,20)(12,19)(14,23)(15,22)(16,21)(18,24)(25,27)(26,33)(28,35)(29,36)(30,38)(31,34)(32,40)(37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759937);