Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
12041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(5,6)(7,8)(10,14)(15,16);(2,4)(9,10)(11,15)(12,16)(13,14);(1,2)(3,4)(6,8)(9,11,13,12)(10,15,14,16);(5,6)(7,8)(9,11,13,12)(10,15,14,16);(2,4)(9,12,13,11)(10,16,14,15);(1,2)(3,4)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi125 conjugacy classes — view the full character table.
| maximal subgroup of | #72238162, #72238205, #72238228, … |
| maximal normal subgroup of | #72238162, #72238205, #72238228, … |
| maximal quotient of | #72257580, #72263844, #72263848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606669);