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 | |
7761 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)(5,6,10,12)(7,11,8,9);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,14)(15,19)(16,17)(18,20);(6,12)(7,8)(14,17)(18,19);(13,14)(15,18)(16,17)(19,20);(5,8,10,7)(6,11,12,9)(13,14)(15,18)(16,17)(19,20);(5,9,10,11)(6,7,12,8)(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238581, #72238584, #72312537, … |
| maximal normal subgroup of | #72238581, #72238584, #72312537, … |
| maximal quotient of | #72261501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7645420);