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 | |
5554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10)(11,12)(13,14)(15,16);(3,7)(6,8)(10,14)(12,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12)(18,22)(20,29)(23,25)(26,30);(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)Polycyclic presentation:
2^9|c^2=g,e^2=h,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 | #72280960, #72281140, #72281311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947816);