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 | |
5496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,20,25,24)(21,29,30,26)(27,31,32,28);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,25)(26,29)(27,32);(2,5)(3,7)(10,14)(11,15)(18,23)(20,24)(21,30)(27,32);(2,5)(6,8)(11,15)(13,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280686 |
| maximal normal subgroup of | #72280686 |
| maximal quotient of | #72280949, #72280982, #72281035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948162);