Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17093 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,4,7,12,6,11)(3,13,17,24,16,28,15,9)(5,18,22,10,21,26,20,27)(14,23,19,25,31,32,30,29);(1,3,6,15,7,16,8,17)(2,9,4,13,12,24,11,28)(5,19,20,14,21,30,22,31)(10,25,26,32,27,29,18,23);(1,4,6,12,7,11,8,2)(3,13,15,24,16,28,17,9)(5,10,20,26,21,27,22,18)(14,23,30,25,31,32,19,29);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,20)(9,23)(11,26)(12,27)(13,29)(15,19)(16,31)(17,30)(24,32)(25,28)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251791, #72251795, … |
| maximal normal subgroup of | #72251786, #72251791, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383801);