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 | |
8191 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)(15,20)(16,19)(17,18);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(15,16)(19,20);(6,9)(10,12)(13,15,18,16)(14,19,17,20);(7,11)(10,12)(15,16)(19,20);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,19,17,20);(5,6)(7,10)(8,9)(11,12)(13,17,18,14)(15,19,16,20)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8005138, #72233991, #72236869, … |
| maximal normal subgroup of | C2×#8005138, #72233991, #72236869, … |
| maximal quotient of | #72233919, #72233920, #72233922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7912334);