Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16839 |
| 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 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(5,19)(8,12)(9,21)(10,22)(13,28)(14,29)(16,31)(24,32)(25,26);(1,2,5,7,6,10,19,22)(3,12,29,20,14,8,11,27)(4,9,18,21,15,26,17,25)(13,30,16,32,31,23,28,24);(1,3,5,11,6,14,19,29)(2,8,7,20,10,12,22,27)(4,16,17,31,15,28,18,13)(9,24,25,30,26,32,21,23);(1,4,5,18,6,15,19,17)(2,9,7,21,10,26,22,25)(3,13,11,28,14,31,29,16)(8,23,20,32,12,30,27,24)Polycyclic presentation:
2^9|b^2=egh,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251792, #72251797, … |
| maximal normal subgroup of | #72251783, #72251792, #72251797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380063);