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 | |
5450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,13,14)(11,12,15,16)(17,18,22,23)(19,26,25,21)(20,24,28,32)(27,30,29,31);(1,2,5,6)(3,7,8,4)(10,14)(11,15)(18,23)(19,25)(20,28)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(18,23)(19,25)(24,32)(27,29);(2,6)(4,7)(9,12)(10,11)(13,16)(14,15)(17,19,22,25)(18,21,23,26)(20,29,28,27)(24,30,32,31);(1,4)(2,3)(5,7)(6,8)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,29)(26,30);(1,3)(2,7)(4,6)(5,8)(17,21)(18,25)(19,23)(20,30)(22,26)(24,27)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72326547 |
| maximal normal subgroup of | #72326547 |
| maximal quotient of | #72326596, #72326711, #72327466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10133154);