Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127645 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,21,25,22,26,37,27)(19,28,31,46,29,45,36,42)(20,32,34,24,35,43,48,44)(23,38,40,30,39,33,41,47);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19)(18,23,26,39)(20,33)(21,36)(22,29)(24,42,44,46)(25,41,27,40)(28,43,45,32)(30,34)(31,37)(35,38)(47,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(19,29)(23,39)(28,45)(30,47)(31,36)(33,38)(40,41)(42,46)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ghi,d^2=g,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,e^b=ei,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240835, #72240836, #72240841, … |
| maximal normal subgroup of | #72240835, #72240836, #72240841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34878);