Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #32890 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18)(8,16)(9,23)(10,27)(11,17)(12,20)(13,19)(14,22)(15,26)(21,34)(24,31)(25,29)(28,33)(30,35)(32,36);(1,2)(3,4)(5,7,11,21)(6,12,17,31)(8,19,10,28)(9,24,26,20)(13,29,33,14)(15,34,23,18)(16,30,27,32)(22,36,25,35);(2,4)(5,7)(6,13)(8,19)(9,24)(10,28)(11,21)(12,29)(14,31)(15,32)(16,34)(17,33)(18,27)(20,26)(22,36)(23,30)(25,35);(5,8,11,10)(6,14,17,29)(7,19,21,28)(9,25,26,22)(12,13,31,33)(15,27,23,16)(18,30,34,32)(20,36,24,35)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72337680, #72340705, #72340715 |
| maximal normal subgroup of | #72337680, #72340705, #72340715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407056);