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 | #42468 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(1,2,5,4)(3,6,7,8)(10,15)(13,17)(16,23)(19,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279396);