Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #234946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35050 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9)(5,18)(6,11)(7,12)(8,14)(10,21)(15,23)(16,29)(17,26)(19,24)(20,27)(22,32)(25,30)(28,31);(1,3,7,16)(2,8,12,25)(4,14,17,30)(5,19,20,32)(6,15,21,31)(9,23,26,28)(10,22,11,24)(13,27,29,18);(4,17)(5,20)(8,22)(9,26)(13,23)(14,30)(18,27)(19,32)(24,25)(28,29);(1,4,6,5)(2,9,11,18)(3,14,15,19)(7,17,21,20)(8,23,24,13)(10,27,12,26)(16,30,31,32)(22,29,25,28)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=di,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239697, #72249937, #72249943, … |
| maximal normal subgroup of | #72239697, #72249937, #72249943, … |
| maximal quotient of | #72257813, #72257816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142148);