Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15527 |
| 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 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,16)(5,20,23,28)(6,11,8,17)(9,25)(10,22,27,24)(14,30)(15,19)(18,32)(21,29)(26,31);(1,3,7,16)(2,9)(4,17)(5,21)(6,15,8,25)(10,18,27,29)(11,13)(12,19)(14,24)(20,30,28,26)(22,31)(23,32);(1,4,8,9,7,13,6,19)(2,3,11,25,12,16,17,15)(5,18,22,26,23,29,24,30)(10,14,28,21,27,31,20,32);(1,5)(2,10)(3,14)(4,18)(6,24)(7,23)(8,22)(9,26)(11,28)(12,27)(13,29)(15,32)(16,31)(17,20)(19,30)(21,25)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319962, #72319982, #72320106, … |
| maximal normal subgroup of | #72319962, #72319982, #72320106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342677);