Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
7881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,10)(7,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(1,2)(3,4)(6,10)(8,11)(15,21)(19,25)(22,27)(26,28);(5,6)(7,8)(9,10)(11,12)(14,18)(19,25)(20,23)(26,28);(5,7)(6,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(5,8,9,11)(6,7,10,12)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28);(5,7)(6,8)(9,12)(10,11)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)Polycyclic presentation:
2^9|a^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72263362, #72297211, #72326930, … |
| maximal normal subgroup of | #72263362, #72297211, #72326930, … |
| maximal quotient of | #72263376, #72297265, #72297494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944769);