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 | #44143 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,18)(11,12,16,15)(13,17,22,24)(19,20,21,23);(2,6)(4,7)(9,11,14,16)(10,15,18,12)(13,21,22,19)(17,20,24,23)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,14,22)(10,17,18,24)(11,19,16,21)(12,20,15,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725440);