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 | |
8225 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)(7,8)(9,11)(10,12)(13,14)(15,19)(16,18)(17,20);(1,2,3,4)(5,7)(6,8)(9,12)(10,11);(5,8,10,9)(6,7,12,11)(13,14,16,18)(15,17,20,19);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20);(5,7)(6,8)(9,12)(10,11)(13,15,16,20)(14,17,18,19);(5,9,10,8)(6,11,12,7)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72315173, #72315179, #72315210 |
| maximal normal subgroup of | #72315173, #72315179, #72315210 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7892434);