Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,19,15,21,31,20)(11,22,25,37,26,38,30,35)(12,17,13,28,27,18,29,36)(16,23,32,24,34,39,33,40);(1,3,5,4)(2,6,7,8);(9,11,15,26)(10,16,21,34)(12,24,27,40)(13,23,29,39)(14,30,31,25)(17,35,18,37)(19,33,20,32)(22,28,38,36);(1,4,5,3)(2,6,7,8)(9,12,14,13,15,27,31,29)(10,17,19,28,21,18,20,36)(11,23,25,24,26,39,30,40)(16,22,32,37,34,38,33,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=i,d^2=eg,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309598, #72309603, #72309607, … |
| maximal normal subgroup of | #72309598, #72309603, #72309607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403204);