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 | |
6514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(17,18)(19,23)(20,21)(22,24);(1,3,6,5)(2,7,4,8)(9,10)(11,12)(13,14)(15,16);(3,5)(7,8)(10,14)(12,15)(18,21)(22,23);(1,2,6,4)(3,8,5,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23);(1,5,6,3)(2,7,4,8)(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233511, #72233536, #72233566, … |
| maximal normal subgroup of | #72233511, #72233536, #72233566, … |
| maximal quotient of | #72261473, #72297516, #72297683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7666832);