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 | |
5758 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)(3,8)(4,7)(5,6)(17,18)(19,25)(20,24)(21,26)(22,23)(27,30)(28,32)(29,31);(1,3,5,7)(2,4,6,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,26)(20,28)(30,31);(9,11)(10,13)(12,16)(14,15)(18,23)(19,26)(24,32)(27,29);(2,6)(3,7)(10,14)(13,15)(17,19)(18,21)(20,29)(22,26)(23,25)(24,30)(27,28)(31,32);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,30)(26,29);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,16)(13,14)(17,21)(18,19)(20,30)(22,25)(23,26)(24,29)(27,32)(28,31)Polycyclic presentation:
2^9|b^2=gh,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 | #72300627, #72342310, #72342323, … |
| maximal normal subgroup of | #72300627, #72342310, #72342323, … |
| maximal quotient of | #72300681, #72300886 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10125032);