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 | |
10125 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)(14,17)(18,19);(1,2)(3,4)(5,6,9,12)(7,8,11,10);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,17)(18,20);(13,15,16,20)(14,18,17,19);(5,7)(6,8)(9,11)(10,12);(5,8)(6,11)(7,12)(9,10)Polycyclic presentation:
2^9|b^2=i,d^2=h,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233745, #72233746, #72233754, … |
| maximal normal subgroup of | #72233745, #72233746, #72233754, … |
| maximal quotient of | #72234038, #72261122, #72261137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591676);