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 | |
5735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(9,10,13,16)(11,12,14,15)(17,18)(19,22)(20,25)(21,24)(23,27)(26,30)(28,32)(29,31);(2,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18,22,19)(20,23,27,25)(21,26,30,24)(28,32,29,31);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,16)(13,15)(18,19)(20,27)(24,26)(28,29);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,16)(13,14)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31);(1,4,5,3)(2,7,6,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,16)(13,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72336231 |
| maximal normal subgroup of | #72336231 |
| maximal quotient of | #72336478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875476);