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 | |
5598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24);(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(18,21)(22,23);(2,5)(3,7)(9,11,14,16)(10,12,15,13)(17,18)(19,22)(20,21)(23,24);(3,7)(6,8)(9,12,14,13)(10,11,15,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,15);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,16,15,11)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72255696, #72263031, #72263040 |
| maximal normal subgroup of | #72255696, #72263031, #72263040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8843895);