Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3274 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)(5,6,8,12,10,15,22,14)(7,16,19,30,20,33,17,32)(9,23,21,13,25,29,34,31)(11,24,27,18,28,35,26,36);(5,7)(6,11)(8,17)(9,24)(10,20)(12,26)(13,30)(14,27)(15,28)(16,29)(18,34)(19,22)(21,36)(23,33)(25,35)(31,32);(2,4);(5,8,10,22)(6,12,15,14)(7,17,20,19)(9,21,25,34)(11,26,28,27)(13,29,31,23)(16,32,33,30)(18,24,36,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239826, #72239835, #72239838, … |
| maximal normal subgroup of | #72239826, #72239835, #72239838, … |
| maximal quotient of | #72309889, #72309913, #72309943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400723);