Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360529 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13877 |
| 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 | |
2237 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)(3,11,14,7)(4,15)(5,19)(6,9)(8,17)(10,21)(12,27,16,29)(13,23,20,25)(18,30)(22,28,26,31)(24,32);(1,3)(2,7)(4,16)(5,20)(6,14)(8,22)(9,11)(10,25)(12,18)(13,21)(15,27)(17,28)(19,23)(24,26)(29,30)(31,32);(1,4,5,17,6,18,21,32)(2,8,10,15,9,24,19,30)(3,12,13,28,14,16,20,31)(7,22,23,29,11,26,25,27);(1,3)(2,7)(4,12)(5,13)(6,14)(8,22)(9,11)(10,23)(15,29)(16,18)(17,28)(19,25)(20,21)(24,26)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267725);