Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24033 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61063 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2930 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
975 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)(4,16)(5,21)(6,11)(7,13)(8,17)(9,27)(10,19)(14,23)(15,26)(18,25)(20,28)(22,29)(24,31)(30,32);(3,13)(4,17)(5,19)(7,23)(8,16)(9,28)(10,21)(12,24)(14,31)(15,18)(20,27)(22,32)(25,29)(26,30);(1,3,5,22,6,14,19,15)(2,7,10,30,11,24,21,25)(4,18,17,31,20,32,27,13)(8,26,16,23,9,29,28,12);(1,2)(3,7)(4,9)(5,10)(6,11)(8,20)(12,31)(13,23)(14,24)(15,25)(16,27)(17,28)(18,29)(19,21)(22,30)(26,32)Polycyclic presentation:
2^9|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=gh,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231700, #72250066, … |
| maximal normal subgroup of | #72213945, #72231700, #72250066, … |
| maximal quotient of | #72226816, #72239894, #72250672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407354);