Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
8242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(5,7,9,10)(6,8,11,12)(13,15)(14,18)(16,20)(17,19);(5,6)(7,12)(8,10)(9,11)(13,16)(14,15)(17,18)(19,20);(13,15)(14,18)(16,20)(17,19);(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,20)(17,18)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72232293, #72234901, #72234916, … |
| maximal normal subgroup of | #72232293, #72234901, #72234916, … |
| maximal quotient of | #72232414, #72232428, #72260775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7764302);