Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13873 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1899 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)(3,12,16,7)(4,17,19,9)(5,11)(6,10)(8,18,26,20)(13,28)(14,23)(15,29,32,24)(21,27)(22,30)(25,31);(1,3,5,15,6,16,21,32)(2,7,11,24,10,12,27,29)(4,14,18,31,19,13,20,30)(8,25,17,28,26,22,9,23);(1,4)(2,8)(3,13)(5,20)(6,19)(7,22)(9,11)(10,26)(12,25)(14,16)(15,31)(17,27)(18,21)(23,29)(24,28)(30,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,23)(8,11)(10,17)(12,28)(13,16)(15,31)(20,21)(22,29)(24,25)(26,27)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72219686, #72219691, … |
| maximal normal subgroup of | #72215906, #72219686, #72219691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267484);