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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,2,5,6)(3,7,4,8)(9,11,14,20)(10,15,18,19)(12,21,22,13)(16,17,23,24)(25,26,31,30)(27,32,29,34)(28,33,38,37)(35,39,36,40);(11,20)(13,21)(15,19)(17,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,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, 7723051);