Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,24)(22,23);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,17,24)(12,21,13,20)(16,23,18,19);(10,16)(12,13)(15,23)(17,18)(19,24)(20,21);(10,17)(15,24)(16,18)(19,23);(2,6)(5,7)(10,17)(15,24)(16,18)(19,23)Polycyclic presentation:
2^9|b^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238938, #72238966, #72249818, … |
| maximal normal subgroup of | #72238938, #72238966, #72249818, … |
| maximal quotient of | #72257861 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256169);