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 | |
5449 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)(3,6)(4,7)(5,8)(10,13)(14,16)(18,22)(19,25)(24,27)(26,28);(1,3)(2,4)(5,7)(6,8)(11,15)(14,16)(18,22)(23,30)(24,27)(31,32);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31);(1,4,5,6)(2,7,8,3)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)Polycyclic presentation:
2^9|f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72327258, #72333432 |
| maximal normal subgroup of | #72327258, #72333432 |
| maximal quotient of | #72327381, #72327815, #72328278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8589437);