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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,18)(12,22)(13,20)(14,16)(15,21)(17,23)(19,24)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(2,3)(6,7)(9,11)(10,15)(12,20)(13,17)(14,21)(16,18)(19,23)(22,24)(27,33)(31,34)(35,37)(39,40);(1,3,5,2)(4,7,8,6)(10,16)(12,23)(15,18)(19,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,12)(10,17)(11,20)(13,15)(14,23)(16,22)(18,24)(19,21)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(9,11)(10,18)(12,20)(13,22)(14,21)(15,16)(17,24)(19,23)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(9,13,14,24)(10,19,16,20)(11,17,21,22)(12,15,23,18)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722391);