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 | |
9134 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)(7,12)(8,10)(9,11)(13,14)(15,16)(17,18)(19,20);(1,2,3,4)(5,7,9,8)(6,10,11,12);(7,8)(10,12)(14,18)(15,19);(5,6)(7,12)(8,10)(9,11)(13,15,17,19)(14,16,18,20);(5,6)(7,12)(8,10)(9,11)(13,16)(14,19)(15,18)(17,20);(5,8,9,7)(6,10,11,12)Polycyclic presentation:
2^9|b^2=g,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233745, #72233766, #72234449, … |
| maximal normal subgroup of | #72233745, #72233766, #72234449, … |
| maximal quotient of | #72261614, #72262064, #72297201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7666923);