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 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6018 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)(3,4,7,8)(9,10,13,15)(11,16,14,12)(20,28)(24,29)(26,32)(30,31);(9,11)(10,12)(13,14)(15,16)(18,22)(23,25)(24,29)(30,31);(2,6)(3,7)(9,12,13,16)(10,14,15,11)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238888, #72285450, #72285461, … |
| maximal normal subgroup of | #72238888, #72285450, #72285461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619859);