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 | #47948 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(27,29)(32,34)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,16)(14,20)(17,23)(18,19)(22,24)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(11,20)(13,24)(15,19)(17,21)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,10)(11,15)(12,22)(13,17)(14,18)(16,23)(19,20)(21,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,12)(10,16)(11,21)(13,15)(14,23)(17,20)(18,22)(19,24)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,22)(12,15)(14,24)(16,20)(18,21)(19,23)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7749636);