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 | |
6348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,20,26,23)(21,25,30,32)(27,28,29,31);(1,3)(2,4)(5,8)(6,7)(17,19)(18,23)(20,24)(21,29)(22,26)(25,28)(27,30)(31,32);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(18,24)(19,26)(21,30)(28,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)Polycyclic presentation:
2^9|a^2=g,c^2=gh,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 | #72261925, #72262117, #72284351, … |
| maximal normal subgroup of | #72261925, #72262117, #72284351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713984);