Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16309 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,6)(9,13)(10,25)(14,30)(15,16)(18,19)(21,24)(22,27)(23,26);(1,2,5,20,7,8,6,11)(3,13,12,21,16,24,15,9)(4,10,17,27,19,22,18,25)(14,29,28,26,30,32,31,23);(1,3,5,12,7,16,6,15)(2,9,20,13,8,21,11,24)(4,14,17,28,19,30,18,31)(10,26,27,32,22,23,25,29);(1,4)(2,10)(3,14)(5,17)(6,18)(7,19)(8,22)(9,23)(11,25)(12,28)(13,29)(15,31)(16,30)(20,27)(21,26)(24,32)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365268);