Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16,19,14)(12,20)(13,17)(15,22)(18,23,21,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,21,22,18)(13,19)(15,24,20,23)(16,17);(2,6)(5,8)(9,12)(10,15,17,20)(11,18)(13,22)(14,23,16,24)(19,21);(3,7)(5,8)(9,11)(10,16,17,14)(12,18)(13,19)(15,23,20,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
| maximal quotient of | #72210069, #72266476, #72266478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387049);