Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15567 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,23,31)(3,13,24,26,16,8,17,30)(4,9,20,28,19,29,25,15)(5,10,27,32,21,18,22,14);(1,3,6,16)(2,8)(4,18,19,10)(5,15)(7,24,23,17)(9,27)(11,30)(12,13)(14,25,32,20)(21,28)(22,29)(26,31);(1,4)(2,9)(3,14)(5,13)(6,19)(7,25)(8,21)(10,17)(11,15)(12,29)(16,32)(18,24)(20,23)(22,26)(27,30)(28,31);(1,5,6,21)(2,10)(3,15)(4,13,19,8)(7,22,23,27)(9,17)(11,14)(12,18)(16,28)(20,30,25,26)(24,29)(31,32)Polycyclic presentation:
2^9|a^2=eg,b^2=f,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318894, #72318910, #72318969, … |
| maximal normal subgroup of | #72318894, #72318910, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343875);