Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14729 |
| 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 | |
1335 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)(3,13,17,24,16,26,15,9)(5,19,20,18,21,10,22,25)(14,28,29,32,31,27,30,23);(1,3,6,15,7,16,8,17)(2,9,4,13,12,24,11,26)(5,14,20,29,21,31,22,30)(10,23,18,27,19,32,25,28);(1,4,6,12,7,11,8,2)(3,9,15,13,16,24,17,26)(5,18,20,19,21,25,22,10)(14,23,29,27,31,32,30,28);(1,5)(2,10)(3,14)(4,18)(6,20)(7,21)(8,22)(9,23)(11,25)(12,19)(13,27)(15,29)(16,31)(17,30)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72215931, #72215938, … |
| maximal normal subgroup of | #72215906, #72215931, #72215938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303148);