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 | |
8836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(6,10)(8,11)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(6,10)(7,12)(14,18)(15,19);(13,15,17,19)(14,16,18,20);(5,7)(6,11)(8,10)(9,12)(13,16)(14,19)(15,18)(17,20);(5,8,9,11)(6,7,10,12)Polycyclic presentation:
2^9|a^2=gi,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233751, #72233754, #72234452, … |
| maximal normal subgroup of | #72233751, #72233754, #72234452, … |
| maximal quotient of | #72257903, #72261665, #72261919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627557);