Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511427 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32554 |
| 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 | |
411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,10,5,4,6,12)(3,13,16,9,17,25,20,15)(7,11,23,26,19,18,21,27)(14,29,22,24,30,28,32,31);(3,14)(6,8)(7,21)(9,24)(10,12)(11,27)(13,28)(15,31)(16,32)(17,30)(18,26)(19,23)(20,22)(25,29);(1,3)(2,9)(4,15)(5,17)(6,20)(7,22)(8,16)(10,25)(11,28)(12,13)(14,21)(18,29)(19,32)(23,30)(24,27)(26,31);(1,4,8,12,5,2,6,10)(3,15,16,13,17,9,20,25)(7,11,23,26,19,18,21,27)(14,24,22,28,30,31,32,29)Polycyclic presentation:
2^9|a^2=ef,d^2=ef,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#511427, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#511427, #72320520, … |
| maximal quotient of | C2×#511427, #72250417, #72250420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418623);