Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
7637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(17,18)(19,20)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,21)(20,23)(22,24);(2,7)(6,8)(10,13)(14,15)(17,20,22,21)(18,19,23,24);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,14,13,15)(17,21,22,20)(18,19,23,24);(1,4,5,3)(2,8,7,6);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72241229, #72241231, #72241232, … |
| maximal normal subgroup of | #72241229, #72241231, #72241232, … |
| maximal quotient of | #72297539, #72297665, #72335303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092535);