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 | |
5872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,4,7,8)(9,10,13,16)(11,12,15,14)(17,18)(19,23)(20,21)(22,24);(2,6)(3,7)(9,11)(10,14)(12,16)(13,15)(18,21)(22,23);(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24);(1,2,5,6)(3,4,7,8)(9,12,13,14)(10,15,16,11)(17,19,20,24)(18,22,21,23);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,15)(13,16)Polycyclic presentation:
2^9|a^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72232817, #72326934, #72336175, … |
| maximal normal subgroup of | #72232817, #72326934, #72336175, … |
| maximal quotient of | #72232961, #72327048, #72327873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9843073);