Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
23162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(14,22)(18,20)(19,21)(23,24);(5,6)(7,8)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24);(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72302893, #72302895, #72303001, … |
| maximal normal subgroup of | #72302893, #72302895, #72303001, … |
| maximal quotient of | #72310488, #72310687, #72313385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538394);