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 | |
6056 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)(3,4)(5,6)(7,8)(17,18)(19,24)(20,30)(21,26)(22,27)(23,29)(25,28)(31,32);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,24)(20,21,28,31)(25,26,30,32);(2,6)(4,8)(10,13)(14,16)(17,19,22,29)(18,24,27,23)(20,21,28,31)(25,32,30,26);(2,6)(3,7)(11,15)(14,16)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,28)(20,29)(22,31)(23,30)(24,25)(27,32);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,30)(31,32)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72235981, #72240625, #72240636, … |
| maximal normal subgroup of | #72235981, #72240625, #72240636, … |
| maximal quotient of | #72235985, #72236001, #72263328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8806480);