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 | #43360 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26)(27,36)(28,29)(30,35)(31,33)(32,37)(34,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27)(26,32)(28,38)(29,35)(30,34)(31,37)(33,36)(39,40);(1,3,5,4)(2,7,6,8)(10,16)(15,18)(17,20)(23,24)(26,33)(27,37)(28,34)(35,40);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28,31,34)(26,29,33,39)(27,38,37,30)(32,35,36,40);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,29)(26,34)(27,35)(28,33)(30,32)(31,39)(36,38)(37,40);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,30,31,38)(26,35,33,40)(27,34,37,28)(29,36,39,32)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7772137);