Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
5332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,25)(20,22)(21,24)(23,26)(27,31)(28,29)(30,32);(3,7)(6,8)(10,13)(14,15)(18,20)(21,29)(23,25)(27,32);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,26)(23,25)(27,32)(30,31);(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,3,5,7)(2,6,4,8)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,4,5,2)(3,8,7,6)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312133, #72318202, #72318207, … |
| maximal normal subgroup of | #72312133, #72318202, #72318207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663301);