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 | |
10006 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)(13,14)(15,19)(16,17)(18,20);(1,2)(3,4)(5,6,9,10)(7,12,8,11)(13,14,16,17)(15,19,20,18);(6,10)(11,12)(14,17)(18,19);(5,7,9,8)(6,11,10,12)(13,14)(15,18)(16,17)(19,20);(5,8,9,7)(6,12,10,11)(13,15,16,20)(14,18,17,19);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233505, #72233538, #72233543, … |
| maximal normal subgroup of | #72233505, #72233538, #72233543, … |
| maximal quotient of | #72261071, #72261185, #72261201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713699);