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 | #55103 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,21)(13,16)(14,20)(15,17)(19,23)(22,24)(26,31)(29,32)(33,39)(37,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,22)(13,19)(15,20)(16,24)(17,18)(21,23);(2,5)(6,7)(9,12,15,13)(10,16,17,21)(11,19,20,22)(14,23,18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(2,5)(6,7)(9,11)(10,14)(12,19)(13,22)(15,20)(16,23)(17,18)(21,24)(25,27,30,34)(26,32,31,29)(28,36,38,35)(33,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,16,17,21)(11,19,20,22)(14,24,18,23)(25,28,30,38)(26,33,31,39)(27,35,34,36)(29,37,32,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,23)(13,24)(15,18)(16,19)(17,20)(21,22)(25,29,30,32)(26,34,31,27)(28,37,38,40)(33,36,39,35)Polycyclic presentation:
2^9|c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231210);