Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94613 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7385 |
| commutator | C32 | C2×C8 |
| Frattini, cosocle | #1397 | C22 |
| prime radical | #2262 | C4 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ #1397 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(2,8,9,31,11,33,13,35)(3,14,17,45)(4,20)(5,25)(6,27)(10,38,36,50,40,56,42,63)(12,44,37,61,43,60,26,34)(15,21,48,30,49,41,32,59)(16,51)(18,52,54,64)(19,58,55,29)(22,28)(23,53)(24,39)(46,62,57,47);(1,2)(3,15)(4,21)(5,26)(6,9)(7,11)(8,29)(10,39)(12,25)(13,27)(14,31)(16,42)(17,49)(18,56)(19,32)(20,40)(22,43)(23,30)(24,41)(28,37)(33,58)(34,62)(35,45)(36,53)(38,54)(44,52)(46,63)(47,61)(48,55)(50,57)(51,59)(60,64)Polycyclic presentation:
2^9|a^2=d,c^2=egh,d^2=f,e^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ceh,d^b=dh,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224652, #72225452, #72225500, … |
| maximal normal subgroup of | #72224652, #72225452, #72225500, … |
| maximal quotient of | #72230702, C2×#94613, C3×#94613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1809);