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 | |
5644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,6,4)(17,18,21,24)(19,22,20,23);(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24);(1,3)(2,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15);(1,4)(2,6)(3,8)(5,7)(9,10)(11,14)(12,15)(13,16);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,19,21,20)(18,22,24,23);(1,3)(2,7)(4,8)(5,6)(9,11)(10,14)(12,16)(13,15)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72234894, #72234895, #72260470, … |
| maximal normal subgroup of | #72234894, #72234895, #72260470, … |
| maximal quotient of | #72260758, #72303527 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566505);