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 | |
6543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,13,14)(11,15,16,12)(18,21)(19,20);(3,8)(6,7)(9,11,13,16)(10,12,14,15)(17,18,22,21)(19,24,20,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,20)(18,23,21,24);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20,22,19)(18,24,21,23);(17,21,22,18)(19,24,20,23);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233520, #72233585, #72234857, … |
| maximal normal subgroup of | #72233520, #72233585, #72234857, … |
| maximal quotient of | #72261302, #72261934, #72297742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378210);