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 | |
5681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,8,6,4)(10,12)(11,13)(17,18,21,19)(20,23,24,22);(1,3)(2,4)(5,6)(7,8)(9,10,14,12)(11,16,13,15)(18,19)(22,23);(1,4)(2,3)(5,8)(6,7)(9,10,14,12)(11,16,13,15)(17,19,21,18)(20,23,24,22);(1,4,5,8)(2,6,7,3)(9,11,14,13)(10,15,12,16)(17,20,21,24)(18,22,19,23);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,15,13,16);(1,3)(2,4)(5,6)(7,8)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297120 |
| maximal normal subgroup of | #72297120 |
| maximal quotient of | #72297130, #72297241, #72297798 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065224);