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 | #53106 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,29)(28,36)(30,32)(31,34)(33,39)(35,40)(37,38);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,30,34)(26,31,32,29)(28,37,39,35)(33,38,36,40);(2,3)(6,7)(13,21)(17,24)(19,20)(22,23)(26,32)(27,34)(33,36)(35,37);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22)(25,28,30,39)(26,33,32,36)(27,35,34,37)(29,38,31,40);(1,3,5,2)(4,7,8,6)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,39)(37,38);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,29,30,31)(26,34,32,27)(28,38,39,40)(33,37,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002156);