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 | |
4309 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)(3,7)(4,5)(6,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(28,36)(32,39)(35,37)(38,40);(3,8)(6,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3)(2,6)(4,7)(5,8)(10,12)(16,18)(17,20)(23,24)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,39,40);(1,2,5,4)(3,7,8,6)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,29,30,27)(26,31,33,34)(28,37,36,35)(32,38,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=gi,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 | #72219281, #72219283 |
| maximal quotient of | #72219276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620966);