Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
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 | |
6734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,8,4,7)(17,18,23,24)(19,21,22,20);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,15,16,14)(17,19,23,22)(18,20,24,21);(2,5)(7,8)(10,12)(14,15)(17,20,23,21)(18,19,24,22);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,12,15)(17,21,23,20)(18,19,24,22);(1,4,6,3)(2,8,5,7);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,15,16,14)(17,22,23,19)(18,20,24,21)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=h,f^2=gi,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 | #72233564, #72233608, #72260885, … |
| maximal normal subgroup of | #72233564, #72233608, #72260885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104157);