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 | |
12346 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)(8,12)(10,11)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14,18,16)(15,19,20,17);(6,7)(10,11)(14,16)(15,20);(13,15,18,20)(14,19,16,17);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,20)(18,19)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^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 | #72216041, #72232033, #72232065, … |
| maximal normal subgroup of | #72216041, #72232033, #72232065, … |
| maximal quotient of | #72232060, #72232150, #72232181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618397);