Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380266 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14262 |
| 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 | |
1907 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)(5,18)(6,9)(8,17)(10,19)(12,25,26,31)(13,21,28,24)(16,29)(20,27,22,32)(23,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,22)(9,11)(10,21)(15,25)(16,26)(17,27)(18,24)(19,28)(20,23)(29,31)(30,32);(1,4,5,17,6,16,19,30)(2,8,10,15,9,23,18,29)(3,12,13,27,14,26,28,32)(7,20,21,31,11,22,24,25);(4,16)(5,19)(10,18)(12,26)(13,28)(15,29)(21,24)(25,31)Polycyclic presentation:
2^9|a^2=h,c^2=f,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287462);