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 | C4 | #1936 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10821 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,13,7,32)(3,14,17,46,20,36,19,52)(4,21,25,55,27,40,26,59)(5,28,29,43,31,60,30,11)(9,33,35,18,37,51,45,16)(10,38,41,23,42,58,54,24)(15,39,49,62,50,64,22,34)(44,56,53,47,61,57,63,48);(1,3,5,18,6,17,29,51,8,20,31,16,7,19,30,33)(2,9,11,36,32,45,60,46,13,37,43,14,12,35,28,52)(4,22,24,56,25,15,38,47,27,49,23,57,26,50,58,48)(10,39,21,53,54,34,59,44,42,64,40,63,41,62,55,61);(1,4,5,23,6,25,29,58,8,27,31,24,7,26,30,38)(2,10,11,40,32,54,60,55,13,42,43,21,12,41,28,59)(3,15,16,47,17,49,33,57,20,50,18,48,19,22,51,56)(9,34,14,44,45,64,52,63,37,62,36,61,35,39,46,53)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314225 |
| maximal normal subgroup of | #72314225 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54137);