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 | |
24016 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)(5,6,7,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24);(5,6)(7,8)(9,11)(10,15)(12,18)(13,21)(14,16)(17,20)(19,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19);(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24);(9,13,14,22)(10,17,18,24)(11,19,16,21)(12,20,15,23)Polycyclic presentation:
2^9|a^2=gi,c^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72234475, #72234478, #72315333, … |
| maximal normal subgroup of | #72234475, #72234478, #72315333, … |
| maximal quotient of | #72261945, #72261970, #72262027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540347);