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 |
| prime radical | #23902 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2384 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,29)(18,30)(19,33)(20,26)(21,27)(22,28)(23,31)(24,32)(25,34)(35,36);(5,7,10,18)(6,11,14,24)(8,19,21,16)(9,17,22,32)(12,25,27,23)(13,15,28,30)(20,34,35,31)(26,33,36,29);(9,22)(17,32)(20,35)(31,34);(2,4)(5,8,10,21)(6,12,14,27)(7,16,18,19)(9,20,22,35)(11,23,24,25)(13,26,28,36)(15,29,30,33)(17,31,32,34)Polycyclic presentation:
2^9|b^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 | #72281572, #72312704, #72312710, … |
| maximal normal subgroup of | #72281572, #72312704, #72312710, … |
| maximal quotient of | #72281660, #72281691, #72282351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418978);