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 | |
5830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,13)(11,16,15,12)(17,18,22,24)(19,23,21,20);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,12,13,16)(17,19,22,21)(18,23,24,20);(1,2)(3,5)(4,6)(7,8)(17,20)(18,21)(19,24)(22,23);(9,12)(10,15)(11,13)(14,16)(17,19,22,21)(18,23,24,20);(1,2)(3,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,16,15,12)(17,21,22,19)(18,23,24,20)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72326547, #72326783 |
| maximal normal subgroup of | #72326547, #72326783 |
| maximal quotient of | #72326565, #72326740, #72326859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9696310);