Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4126 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)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,24,30)(8,20,21,13,23,28,22,29)(17,31,34,36,33,35,32,25);(2,4)(5,7)(6,12,15,27)(8,17)(9,18)(10,24)(11,19)(13,25,29,36)(14,26,16,30)(20,31,28,35)(21,32)(22,34)(23,33);(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,25)(14,28)(15,29)(16,20)(18,32)(19,33)(24,34)(26,35)(27,36)(30,31);(5,9,11,10)(6,14,15,16)(7,18,19,24)(8,21,23,22)(12,26,27,30)(13,28,29,20)(17,32,33,34)(25,35,36,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237649, #72237659, #72237670, … |
| maximal normal subgroup of | #72237649, #72237659, #72237670, … |
| maximal quotient of | #72257293, #72263946, #72263998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260664);