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 | |
4497 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,30)(28,36)(29,33)(31,35)(32,37)(34,39)(38,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,22,24,17)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,8)(4,7)(5,6)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,16,18,24)(11,13,21,23)(15,22,19,17)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,4,5,3)(2,7,6,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,16)(18,22)(19,24)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,22)(14,18)(17,23)(19,21)(20,24)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72219287 |
| maximal normal subgroup of | #72219283, #72219285, #72219287 |
| maximal quotient of | #72219272 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9636408);