Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467244 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16562 |
| 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 | |
979 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)(3,6)(4,13)(5,10,21,27)(7,11)(8,12)(9,16)(14,20,28,22)(15,26,31,29)(17,25)(18,24)(19,23,30,32);(1,3,7,12,8,11,6,2)(4,9,17,24,18,25,16,13)(5,14,20,27,21,28,22,10)(15,23,19,26,31,32,30,29);(1,4,7,16,8,18,6,17)(2,9,3,13,12,25,11,24)(5,19,20,15,21,30,22,31)(10,26,14,23,27,29,28,32);(1,5)(2,10)(3,14)(4,15)(6,22)(7,20)(8,21)(9,23)(11,28)(12,27)(13,29)(16,30)(17,19)(18,31)(24,26)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251676, #72251686, #72251700, … |
| maximal normal subgroup of | #72251676, #72251686, #72251700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374440);