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 | #40496 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4566 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)(9,10,14,16)(11,17,18,12)(13,15,23,24)(19,22,21,20)(25,26,31,33)(27,32,30,34)(28,29,37,40)(35,36,38,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,21)(14,18)(15,22)(16,17)(19,23)(20,24)(25,27,31,30)(26,32,33,34)(28,35,37,38)(29,36,40,39);(1,3,5,4)(2,7,6,8)(11,18)(12,17)(19,21)(20,22)(26,33)(28,37)(32,34)(35,38);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,29)(27,35)(30,38)(31,37)(32,36)(33,40)(34,39);(9,12)(10,11)(13,22)(14,17)(15,21)(16,18)(19,24)(20,23)(25,29)(26,28)(27,36)(30,39)(31,40)(32,35)(33,37)(34,38);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,15,16,24)(11,19,18,21)(12,20,17,22)(25,30,31,27)(26,34,33,32)(28,38,37,35)(29,39,40,36)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882149);