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 | |
5195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,12,16,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,30)(27,31)(29,32);(1,2,4,6)(3,7,8,5)(9,11,13,16)(10,14,15,12)(18,22)(23,25)(24,28)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(19,26)(23,25)(27,29)(31,32);(1,2,4,6)(3,5,8,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,30)(28,31);(1,2)(3,5)(4,6)(7,8)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29);(1,3,4,8)(2,5,6,7)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72300328 |
| maximal normal subgroup of | #72300328 |
| maximal quotient of | #72300361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8735901);