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 | |
4526 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,28)(27,35,38,32)(29,33,39,37)(30,36,40,34);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,35)(29,34)(30,37)(31,38)(33,40)(36,39);(1,2,5,4)(3,6,8,7)(10,14)(11,18)(16,20)(19,24)(25,26,31,28)(27,35,38,32)(29,37,39,33)(30,34,40,36);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,35,38,32)(29,37,39,33)(30,36,40,34);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,29,31,39)(26,33,28,37)(27,36,38,34)(30,35,40,32);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30,31,40)(26,34,28,36)(27,37,38,33)(29,35,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625603);