Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,13,7,12,5,11)(3,14,16,27,17,8,21,23)(4,9,18,25,19,10,20,26)(15,22,30,32,29,24,28,31);(1,3)(2,8)(4,15)(5,16)(6,21)(7,17)(9,24)(10,22)(11,23)(12,14)(13,27)(18,28)(19,29)(20,30)(25,32)(26,31);(1,4)(2,9)(3,15)(5,20)(6,18)(7,19)(8,22)(10,12)(11,26)(13,25)(14,24)(16,30)(17,29)(21,28)(23,32)(27,31);(1,4)(2,10)(3,15)(5,18)(6,20)(7,19)(8,22)(9,12)(11,26)(13,25)(14,24)(16,28)(17,29)(21,30)(23,31)(27,32);(1,5,7,6)(2,11,12,13)(3,16,17,21)(4,18,19,20)(8,23,14,27)(9,25,10,26)(15,28,29,30)(22,31,24,32)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,e^c=ei,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72310760, #72310766, #72310784, … |
| maximal normal subgroup of | #72310760, #72310766, #72310784, … |
| maximal quotient of | #72318704 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281393);