Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16285 |
| 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 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,20,7,9,6,11)(3,12,13,23,16,21,15,8)(4,10,19,26,18,24,17,25)(14,29,28,22,30,27,31,32);(1,3,5,13,7,16,6,15)(2,8,20,12,9,23,11,21)(4,14,19,28,18,30,17,31)(10,22,26,27,24,32,25,29);(2,9)(3,13)(4,17)(5,6)(8,21)(10,25)(12,23)(14,30)(15,16)(18,19)(22,32)(24,26);(1,4)(2,10)(3,14)(5,19)(6,17)(7,18)(8,22)(9,24)(11,25)(12,27)(13,28)(15,31)(16,30)(20,26)(21,29)(23,32)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363932);