Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3543 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(6,8)(10,11)(13,14)(15,19)(16,17)(18,20);(1,2,4,3)(14,17)(18,19);(5,6,9,8)(7,10,12,11)(13,14)(15,18)(16,17)(19,20);(5,7,9,12)(6,10,8,11)(13,15,16,20)(14,18,17,19);(1,3,4,2)(5,7)(6,10)(8,11)(9,12);(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh146 conjugacy classes — view the full character table.
| maximal subgroup of | #72238156, #72238161, #72238202, … |
| maximal normal subgroup of | #72238156, #72238161, #72238202, … |
| maximal quotient of | #72263726, #72263750, #72263791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534358);