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 | |
8880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(15,19)(18,20);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,14,17,16)(15,18,19,20);(6,12)(7,8)(13,15)(14,18)(16,20)(17,19);(5,7,10,8)(6,11,12,9)(13,16,17,14)(15,18,19,20);(5,8,10,7)(6,9,12,11);(5,9)(6,7)(8,12)(10,11)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72304429, #72304462, #72304609, … |
| maximal normal subgroup of | #72304429, #72304462, #72304609, … |
| maximal quotient of | #72305021, #72306012, #72313718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578787);