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 | #53181 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(5,7)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26)(27,35)(28,31)(29,38)(30,36)(32,37)(33,39)(34,40);(2,4)(3,8)(10,15)(12,16)(17,21)(20,23)(26,28)(29,39)(32,35)(34,36);(1,2,6,4)(3,7,8,5)(25,27)(26,32)(28,35)(29,36)(30,33)(31,37)(34,39)(38,40);(1,3,6,8)(2,7,4,5)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,28,31,26)(27,35,37,32)(29,38,39,33)(30,36,40,34);(1,4,6,2)(3,5,8,7)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29)(26,33)(27,36)(28,38)(30,32)(31,39)(34,37)(35,40);(1,5)(2,3)(4,8)(6,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30,31,40)(26,34,28,36)(27,33,37,38)(29,32,39,35)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219281, #72219283 |
| maximal quotient of | #72219274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015949);