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 | |
23002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(5,6,7,8);(5,6)(7,8)(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,23)(21,22);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23);(9,12,15,11)(10,16,18,19)(13,20,17,22)(14,23,24,21);(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23);(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^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 | #72249643, #72256860, #72262356, … |
| maximal normal subgroup of | #72249643, #72256860, #72262356, … |
| maximal quotient of | #72264109, #72310604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539666);