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 | #59307 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,12,17,20)(13,16,22,21)(18,19,23,24);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,21)(14,15)(16,22)(17,20)(18,24)(19,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,35)(36,40)(37,38);(1,2,5,4)(3,6,8,7)(10,15)(11,17)(13,22)(19,24)(26,29)(31,34)(32,37)(39,40);(9,11,14,17)(10,12,15,20)(13,18,22,23)(16,19,21,24)(25,27,30,35)(26,31,29,34)(28,36,38,33)(32,40,37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24)(25,28,30,38)(26,32,29,37)(27,33,35,36)(31,39,34,40);(9,13,14,22)(10,16,15,21)(11,18,17,23)(12,19,20,24)(25,29,30,26)(27,34,35,31)(28,37,38,32)(33,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=i,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, 7756215);