Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1950 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)(5,6,10,16,11,17,8,14)(7,18,22,12,19,28,20,27)(9,13,23,30,24,15,25,29)(21,32,35,26,34,33,36,31);(1,2,3,4)(5,7)(6,12)(8,22)(9,21)(10,20)(11,19)(13,26)(14,28)(15,31)(16,18)(17,27)(23,36)(24,34)(25,35)(29,33)(30,32);(6,13)(8,10)(9,24)(12,26)(14,29)(15,17)(16,30)(18,32)(20,22)(21,34)(27,31)(28,33);(7,19)(12,27)(18,28)(20,22)(21,34)(26,31)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=gi,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 | #72266731, #72267264, #72267625, … |
| maximal normal subgroup of | #72266731, #72267264, #72267625, … |
| maximal quotient of | #72266734, #72266764, #72267275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401813);