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 |
| prime radical | #2404 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4441 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26,30,35)(27,32,36,31)(28,33,38,39)(29,34,40,37);(1,2)(3,7)(4,8)(5,6)(25,27)(26,31)(28,37)(29,39)(30,36)(32,35)(33,40)(34,38);(10,15)(11,17)(16,22)(18,23)(27,36)(28,38)(31,32)(33,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,27)(26,32)(28,37)(29,33)(30,36)(31,35)(34,38)(39,40);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28,30,38)(26,33,35,39)(27,37,36,34)(29,31,40,32);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29)(26,34)(27,33)(28,31)(30,40)(32,38)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643748);