Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15924 |
| 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 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,16,9,17,25,22,24)(5,14,19,26,21,18,20,10)(15,23,29,27,31,28,30,32);(1,3,6,11,8,12,7,2)(4,9,16,13,17,24,22,25)(5,18,19,26,21,14,20,10)(15,28,29,27,31,23,30,32);(1,4)(2,9,11,24)(3,13,12,25)(5,15)(6,16)(7,22)(8,17)(10,23,26,28)(14,27,18,32)(19,29)(20,30)(21,31);(1,5,6,20,8,21,7,19)(2,10,3,14,11,26,12,18)(4,15,16,30,17,31,22,29)(9,23,13,27,24,28,25,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=eg,f^2=i,g^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=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251789, #72251796, … |
| maximal normal subgroup of | #72251786, #72251789, #72251796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356087);