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 | |
5391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,21)(19,26,22,24)(20,28,30,25)(27,31,29,32);(1,2)(3,6)(4,7)(5,8)(17,19)(18,24)(20,29)(21,26)(22,23)(25,32)(27,30)(28,31);(9,10,12,14)(11,13,16,15)(19,22)(24,26)(27,29)(31,32);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32);(1,4,5,3)(2,7,8,6)(17,22,23,19)(18,24,21,26)(20,29,30,27)(25,31,28,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326547 |
| maximal normal subgroup of | #72326547 |
| maximal quotient of | #72326549, #72326723, #72327374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7714014);