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 | |
4646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,4,7,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,31)(28,35)(29,37)(30,33)(32,38)(34,36)(39,40);(1,3)(2,6)(4,8)(5,7)(11,17)(15,21)(18,23)(22,24)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(10,14)(15,21)(16,19)(22,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3)(2,4)(5,7)(6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,28)(29,37)(30,33)(31,35)(32,38)(34,39)(36,40);(1,4)(2,7)(3,8)(5,6)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=hi,f^2=hi,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 normal subgroup of | #72219279, #72219281, #72219285 |
| maximal quotient of | #72219273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9634170);