Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452545 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| 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 | |
917 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)(3,10,15,8)(4,16)(5,22,23,25)(6,13)(7,12)(9,19)(11,18)(14,26,30,32)(17,24,31,27)(20,28)(21,29);(1,3,7,5,6,15,18,23)(2,8,11,22,13,10,12,25)(4,17,19,14,21,31,20,30)(9,26,16,24,28,32,29,27);(1,4,18,19,6,21,7,20)(2,9,11,29,13,28,12,16)(3,14,5,17,15,30,23,31)(8,24,25,32,10,27,22,26);(1,5,18,15,6,23,7,3)(2,10,11,22,13,8,12,25)(4,17,19,30,21,31,20,14)(9,27,29,26,28,24,16,32)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251695, #72251710, … |
| maximal normal subgroup of | #72251686, #72251695, #72251710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359741);