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 | |
9936 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)(5,6,9,10)(7,8,11,12);(1,2,3,4)(13,14)(15,18)(16,17)(19,20);(6,10)(8,12)(14,17)(18,20);(6,10)(7,11)(15,19)(18,20);(5,7,9,11)(6,8,10,12)(13,15,16,19)(14,18,17,20);(5,8)(6,11)(7,10)(9,12)(13,14)(15,18)(16,17)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233761, #72233764, #72234465, … |
| maximal normal subgroup of | #72233761, #72233764, #72234465, … |
| maximal quotient of | #72256739, #72256781, #72262137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7557181);