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 | #38439 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(13,22)(16,23)(17,20)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,23)(22,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(10,15)(12,21)(16,23)(18,24)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,23)(12,18,21,24)(13,17,22,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,15,20)(11,18,19,24)(13,16,22,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,13,14,22)(10,18,15,24)(11,17,19,20)(12,16,21,23)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,e^2=g,f^2=hi,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, 7627404);