Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24338 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2961 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12,15,31)(4,17)(5,21)(7,24,25,13)(8,27)(9,18)(10,22)(14,30,26,29)(16,19,23,32)(20,28);(1,2)(3,13,15,24)(4,18)(5,22)(6,11)(7,12,25,31)(8,28)(9,17)(10,21)(14,32,26,19)(16,30,23,29)(20,27);(1,3)(2,7)(4,19)(5,23)(6,15)(8,29)(9,30)(10,14)(11,25)(12,28)(13,27)(16,21)(17,31)(18,24)(20,32)(22,26);(1,2)(3,14,15,26)(4,8)(5,10)(6,11)(7,23,25,16)(9,20)(12,29,31,30)(13,32,24,19)(17,27)(18,28)(21,22)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=e,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, C2×#500940, #72249754, … |
| maximal normal subgroup of | #72213937, C2×#500940, #72249754, … |
| maximal quotient of | #72231475, #72231541, C2×#500940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408136);