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 | |
9252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,18)(15,16,19,20);(1,2)(3,4)(5,6,10,8)(7,12,9,11);(5,7,10,9)(6,11,8,12)(14,18)(15,19);(5,6,10,8)(7,12,9,11)(13,15)(14,16)(17,19)(18,20);(5,8,10,6)(7,12,9,11)(13,16)(14,15)(17,20)(18,19);(5,9,10,7)(6,11,8,12)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,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 | #72233742, #72233747, #72233752, … |
| maximal normal subgroup of | #72233742, #72233747, #72233752, … |
| maximal quotient of | #72234026, #72261463, #72261621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7849371);