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 | #37536 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4636 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)(11,15)(12,18)(13,17)(14,16)(19,22)(20,24)(21,23)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,39)(37,38);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(26,32)(27,35)(28,29)(38,39);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(13,21)(20,22)(25,27,31,35)(26,33,32,40)(28,36,29,37)(30,39,34,38);(9,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,23)(22,24)(25,28,31,29)(26,30,32,34)(27,36,35,37)(33,39,40,38);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,29,31,28)(26,34,32,30)(27,37,35,36)(33,38,40,39);(9,13,14,21)(10,17,16,23)(11,19,18,24)(12,20,15,22)(25,30,31,34)(26,28,32,29)(27,38,35,39)(33,37,40,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670632);