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 | |
5719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,24)(19,21,23,20);(2,5)(6,7)(9,11,13,14)(10,12,15,16)(17,19,22,23)(18,21,24,20);(2,5)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,23)(22,24);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,21,22,20)(18,23,24,19);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72262836, #72262931, #72262996, … |
| maximal normal subgroup of | #72262836, #72262931, #72262996, … |
| maximal quotient of | #72300290, #72300367, #72300872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962631);