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 | |
8234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(5,6,9,8)(7,10,12,11)(13,14)(15,18)(16,17)(19,20);(2,4)(6,8)(10,11);(1,2)(3,4)(5,6,9,8)(7,11,12,10)(14,17)(18,20);(13,14)(15,18)(16,17)(19,20);(2,4)(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,18,17,20);(1,2)(3,4)(5,8,9,6)(7,10,12,11)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi125 conjugacy classes — view the full character table.
| maximal subgroup of | #72234902, #72234905, #72234920, … |
| maximal normal subgroup of | #72234902, #72234905, #72234920, … |
| maximal quotient of | #72261393, #72261753 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606740);