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 | |
8207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(6,9)(10,12)(15,20)(17,19);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20);(6,9)(10,12)(13,15,16,20)(14,17,18,19);(6,9)(7,11)(14,18)(17,19);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,20)(17,18);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,18)(17,20)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72327281, #72327640, #72336994 |
| maximal normal subgroup of | #72327281, #72327640, #72336994 |
| maximal quotient of | #72327355, #72327683, #72327746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988283);