Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #431482 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15439 |
| 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 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,18)(3,12,15,8,13,25,19,21)(4,9,16,26,17,23,20,24)(14,29,31,32,28,27,30,22);(3,13)(6,7)(8,21)(9,23)(10,18)(16,20)(22,32)(30,31);(1,3,7,15,5,13,6,19)(2,8,10,25,11,21,18,12)(4,14,16,31,17,28,20,30)(9,22,26,29,23,32,24,27);(1,4,7,16,5,17,6,20)(2,9,10,26,11,23,18,24)(3,14,15,31,13,28,19,30)(8,22,25,29,21,32,12,27)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338678);