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 | |
6484 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)(10,15)(11,12)(17,18,20,21)(19,23,24,22);(3,8)(6,7)(9,10)(11,16)(12,13)(14,15)(18,21)(22,23);(1,3)(2,6)(4,7)(5,8)(9,11,14,12)(10,16,15,13)(17,18)(19,22)(20,21)(23,24);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,15,13)(17,18)(19,22)(20,21)(23,24);(17,19,20,24)(18,22,21,23);(1,4,5,2)(3,7,8,6)(9,13,14,16)(10,12,15,11)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,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, #72233542, #72233557, … |
| maximal normal subgroup of | #72233520, #72233542, #72233557, … |
| maximal quotient of | #72261919, #72297407, #72297546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378207);