Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,19,21,31)(8,13,20,32)(9,26)(10,28)(11,30)(12,18)(14,22,34,24)(15,27)(16,29)(17,23)(25,35,36,33);(5,7,10,20,11,22,29,36,12,21,9,8,23,24,27,25)(6,13,15,31,17,33,26,34,18,32,16,19,30,35,28,14);(2,4)(5,8,9,24,11,25,27,7,12,20,10,22,23,36,29,21)(6,14,15,32,17,31,26,35,18,34,16,13,30,19,28,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239804, #72239814, #72239831, … |
| maximal normal subgroup of | #72239804, #72239814, #72239831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54132);