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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4644 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)(9,10,15,13)(11,16,14,18)(12,17,21,22)(19,23,20,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,38)(34,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,21)(22,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(11,14)(16,18)(19,20)(23,24)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24)(25,27)(26,31)(28,37)(29,35)(30,33)(32,39)(34,38)(36,40);(1,2,5,4)(3,6,8,7)(9,13,15,10)(11,18,14,16)(12,17,21,22)(19,23,20,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,13,16)(12,20,21,19)(17,24,22,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,e^2=gi,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, 7754132);