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 | |
6293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,18,22,21)(19,20,24,23);(3,7)(5,8)(9,10)(11,12)(13,14)(15,16)(18,21)(19,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,24)(18,23,21,20);(11,15)(12,16)(17,20)(18,19)(21,24)(22,23);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,15,14,11)(17,21,22,18)(19,20,24,23);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=ghi,c^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72232554, #72326546, #72326662, … |
| maximal normal subgroup of | #72232554, #72326546, #72326662, … |
| maximal quotient of | #72232638, #72326592, #72326699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091969);