Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,31)(20,32)(21,27)(22,36)(23,29)(24,30)(25,33)(26,35)(28,34);(5,7,11,20)(6,12,16,26)(8,18,9,22)(10,19,24,35)(13,25,14,28)(15,17,30,32)(21,33,23,34)(27,31,29,36);(5,8,11,9)(6,13,16,14)(7,18,20,22)(10,23,24,21)(12,25,26,28)(15,27,30,29)(17,31,32,36)(19,34,35,33);(2,4)(5,9,11,8)(6,14,16,13)(7,18,20,22)(10,23,24,21)(12,25,26,28)(15,29,30,27)(17,31,32,36)(19,33,35,34)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72233533, #72281572, #72312236, … |
| maximal normal subgroup of | #72233533, #72281572, #72312236, … |
| maximal quotient of | #72233468, #72234965, #72281618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418980);