Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
506 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)(6,13,16,24)(7,18)(8,23)(9,11)(10,25)(14,30,32,34)(15,33,26,17)(19,36,31,29)(20,35)(22,27);(2,4)(5,6,10,17,9,26,25,13,12,16,23,33,11,15,8,24)(7,19,21,29,27,14,20,30,22,31,28,36,18,32,35,34);(5,7)(6,14,16,32)(8,20)(9,27)(10,28)(11,18)(12,22)(13,29,24,36)(15,19,26,31)(17,34,33,30)(21,23)(25,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250420, #72250445, #72250464, … |
| maximal normal subgroup of | #72250420, #72250445, #72250464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54403);