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 | |
8362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12);(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,20,19);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20);(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,17,18,19);(5,7)(6,11)(8,10)(9,12)Polycyclic presentation:
2^9|c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72232192, #72232293, #72234904, … |
| maximal normal subgroup of | #72232192, #72232293, #72234904, … |
| maximal quotient of | #72232208, #72232432, #72232434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809461);