Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500351 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24089 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61259 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 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)(3,12)(4,17)(5,22)(6,11)(7,15)(8,20)(9,27)(10,19)(13,14)(16,26)(18,25)(21,29)(23,28)(24,31)(30,32);(1,2)(3,13,14,12)(4,17)(5,22)(6,11)(7,15,24,31)(8,20)(9,27)(10,19)(16,28,23,26)(18,30,32,25)(21,29);(1,3,5,23,6,14,19,16)(2,7,10,30,11,24,22,25)(4,18,20,31,21,32,27,15)(8,26,17,13,9,28,29,12);(1,2)(3,7)(4,9)(5,10)(6,11)(8,21)(12,31)(13,15)(14,24)(16,25)(17,27)(18,28)(19,22)(20,29)(23,30)(26,32)Polycyclic presentation:
2^9|b^2=e,c^2=eh,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, #72231596, #72231599, … |
| maximal normal subgroup of | #72213937, #72231596, #72231599, … |
| maximal quotient of | #72231484, #72237224, C2×#500351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407547);