Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478598 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17156 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,8,15,25,16,23,14,24)(4,17,21,28,19,27,20,9)(13,29,18,22,30,26,31,32);(1,3)(2,8)(4,18,19,31)(5,14)(6,15)(7,16)(9,26,28,29)(10,25)(11,24)(12,23)(13,21,30,20)(17,32,27,22);(1,4,5,20,7,19,6,21)(2,9,10,17,12,28,11,27)(3,13,15,18,16,30,14,31)(8,22,24,29,23,32,25,26);(2,10)(3,14)(4,19)(5,6)(8,23)(9,27)(11,12)(13,18)(15,16)(17,28)(26,29)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72339676, #72339680, #72339684 |
| maximal normal subgroup of | #72339676, #72339680, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385794);