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 | |
9110 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)(14,17)(18,20);(1,2)(3,4)(6,8)(10,12);(5,6,9,8)(7,10,11,12)(13,14)(15,18)(16,17)(19,20);(7,11)(10,12);(5,7)(6,10)(8,12)(9,11);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|c^2=i,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72233750, #72233756, #72234453, … |
| maximal normal subgroup of | #72233750, #72233756, #72234453, … |
| maximal quotient of | #72261502, #72261526, #72261847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540305);