Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
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 | |
13276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24);(5,6,8,7)(11,18)(15,22)(19,23)(20,24);(1,2,3,4)(5,7,8,6)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,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 | #72234480, #72234486, #72260755, … |
| maximal normal subgroup of | #72234480, #72234486, #72260755, … |
| maximal quotient of | #72261504, #72261533, #72261550 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538819);