Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,19,29)(9,21,23,13)(10,14)(12,20)(15,30,31,35)(16,25)(17,28,33,26)(22,27)(24,32)(34,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,28)(14,25)(18,35)(19,31)(20,32)(21,26)(22,34)(23,33)(27,36)(29,30);(5,8,10,19)(6,12,14,27)(7,15,16,31)(9,22,23,20)(11,24,25,36)(13,29,21,18)(17,34,33,32)(26,35,28,30);(7,16)(11,25)(15,31)(17,33)(24,36)(26,28)(30,35)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256610, #72256625, #72256628, … |
| maximal normal subgroup of | #72256610, #72256625, #72256628, … |
| maximal quotient of | #72263420, #72263422, #72263550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401793);