Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435783 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15538 |
| 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 | |
1039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,8,6,12,5,11)(3,9,13,23,15,24,20,21)(4,16,17,25,18,26,19,10)(14,28,27,22,30,31,29,32);(2,8)(3,13)(4,17)(5,7)(10,25)(11,12)(15,20)(18,19)(21,23)(22,28)(27,29)(31,32);(1,3)(2,9)(4,14)(5,13)(6,15)(7,20)(8,21)(10,22)(11,23)(12,24)(16,28)(17,29)(18,30)(19,27)(25,32)(26,31);(1,4,6,18)(2,10)(3,14,15,30)(5,19,7,17)(8,16)(9,22)(11,26)(12,25)(13,27,20,29)(21,28)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,g^2=i,h^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,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237070, #72237086, … |
| maximal normal subgroup of | #72237064, #72237070, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342979);